/»!**•  ITY\ 

LIBRARY 

UMIVfltSI"  OP     I 


THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 


GIFT  OF 

Prof.  G.  C.  Evans 


ELEMENTS  OF  THE  THEOEY  OF  THE 

NEWTONIAN  POTENTIAL 

FUNCTION 

Third,  Revised  and  Enlarged  Edition 
BY 

B.  O.  PEIRCE,  PH.D. 

HOLLIS  PROFESSOR  OF  MATHEMATICS  AND  NATURAL 
PHILOSOPHY  IN  HARVARD  UNIVERSITY 


BOSTON,  U.S.A. 

GINN  &  COMPANY,  PUBLISHERS 
bcnatum  J)tc00 
1902 


COPYRIGHT,  1886,  1902,  BY 
GINN  &  COMPANY 


ALL  BIGHTS   RESERVED 


/MATH/ 
ST/AT 


PREFACE  TO  THIRD  EDITION. 

THIS  little  text-book  was  written  some  years  ago  to  accom 
pany  the  lectures  in  a  short  preparatory  course  on  the  New 
tonian  Potential  Function,  especially  intended  for  students 
who  were  afterwards  to  begin  a  systematic  study  of  the 
Mathematical  Theory  of  Electricity  and  Magnetism,  with  the 
help  of  some  of  the  standard  treatises  on  the  subject. 

In  preparing  the  present  edition  a  few  imperative  changes 
have  been  made  in  the  plates,  some  sections  have  been  intro 
duced,  and  a  large  number  of  simple  miscellaneous  problems 
have  been  added  at  the  end  of  the  last  chapter. 

The  reader  who  wishes  to  get  a  thorough  knowledge  of 
the  properties  of  the  Potential  Function  and  of  its  appli 
cations,  is  referred  to  the  works  mentioned  in  the  list  given 
below.  Most  of  those  that  had  then  been  published  I  con 
sulted  and  used  in  writing  these  notes,  and  from  some  which 
have  appeared  since  the  body  of  this  book  was  electro- 
typed  I  have  borrowed  material  for  problems  :  many  other 
problems  I  have  taken  from  various  college  and  university 
examination  papers.  I  am  indebted  also  to  my  colleagues, 
Professors  Trowbridge,  Byerly,  E.  H.  Hall,  Osgood,  Sabine, 
M.  Bocher,  and  C.  A.  Adams  for  valuable  criticisms  and 
suggestions. 

The  slight  use  which  I  have  made  of  developments  in  terms 
of  Spherical  Harmonics  and  Bessel's  Functions  is  explained 
by  the  fact  that  students  who  use  this  book  in  Harvard  Uni 
versity  study  at  the  same  time  Professor  Byerly's  admirable 
Treatise  on  Fourier's  Series,  and  Spherical,  Cylindrical  dhid 
Ell  ipso  Ida  I  Ha  nn  on  ics. 

iii 


313 


IV  PREFACE    TO    THIRD    EDITION. 

In  the  following  pages  the  change  made  in  a  function  u 
by  giving  to  the  independent  variable  x  the  arbitrary  incre 
ment  A#,  and  keeping  the  other  independent  variables,  if 
there  are  any,  unchanged,  is  denoted  by  Axw.  Similarly, 
&yu  and  AgW  represent  the  increments  of  u  due  to  changes 
respectively  in  y  alone  and  in  z  alone.  The  total  change  in 
u  due  to  simultaneous  changes  in  all  the  independent  variables 
is  sometimes  denoted  by  Aw ;  so  that,  if  u  =f(x,  y,  z), 

&xu  \u  &zu 

Aw  =  -—•  •  Ax  +  -*—  -  A?/  +  -  -  '  &z  +  c, 
Ax  Ay  Az 

where  e  is  an  infinitesimal  of  an  order  higher  than  the  first. 

The  partial  derivatives,  -5-,  -r-,  -T-,  are  denoted,  for  conven- 

dx   dy    dz 

ience,  by  Dxu,  Dyu,  and  Dzu,  and  the  sign  =  placed  between 
a  variable  and  a  constant  is  used  to  show  that  the  former  is 
to  be  made  to  approach  the  latter  as  its  limit.  In  those  cases 
where  it  is  desirable  to  draw  attention  to  the  fact  that  a  cer 
tain  derivative  is  total,  the  differential  notation  —  is  used. 

dx 

It  is  tacitly  assumed  that  the  physical  quantities  under  con 
sideration  can  be  represented  in  the  regions  to  which  the 
theorems  refer,  by  continuous  point  functions,  having  con 
tinuous  derivatives  of  the  orders  which  present  themselves  in 
the  investigation  in  hand.  In  a  few  instances,  as  the  reader 
will  see,  a  theorem  is  predicated  of  analytic  functions  only, 
when  so  narrow  a  limitation  is  not  required  by  the  proof 
given. 


SHORT   LIST   OF   WORKS   ON   THE   POTENTIAL 
FUNCTION   AND   ITS   APPLICATIONS. 

Bacharach :  Abriss  der  Geschichte  der  Potentialtheorie. 

Bedell  and  Crehore :  Alternating  Currents. 

Betti :  Teorica    delle    Forze    Newtoniane    e    sue    Applicazioni    all' 

Elettrostatica  e  al  Magnetismo.     Also  W.  F.  Meyer's  transla 
tion  of  the  same  work  into  German. 
Bocher:  Reihenentwicklungen  der  Potentialtheorie. 
Boltzmann :  Yorlesungen   iiber  Maxwell's  Theorie  der  Elektricitat 

und  des  Lichtes. 

Burkhardt  and  Meyer  :  Die  Potentialtheorie. 
Chrystal:   The   article  "Electricity"  in  the  Ninth   Edition  of  the 

Encyclopaedia  Britannica. 

Clausius  :  Die  Potentialfunction  und  das  Potential. 
Cumming  :   An  Introduction  to  the  Theory  of  Electricity. 
Dirichlet :  Yorlesungen  iiber   die  im  umgekehrten  Verhaltniss  des 

Quadrats     der    Entfernung    wirkenden    Krafte.      Edited    by 

Grube. 

Drude  :  Physik  des  Aethers. 

Duhem  :  Lemons  sur  1'Electricite"  et  le  Magne"tisme. 
Ewing :  Magnetic  Induction  in  Iron  and  other  Metals. 
Ferrers :  Spherical  Harmonics. 
Fleming:   The  Alternate  Current  Transformer. 
Franklin  and  Williamson:  The  Elements  of  Alternating  Currents. 
Gauss:  Allgemeine  Lehrsatze  in  Beziehung  auf  die  im  verkehrten 

Yerhaltnisse  des  Quadrates  der  Entfernung  wirkenden  Anzie- 

hungs-  und  Abstossungskrafte.     Also  other  papers  to  be  found 

in  Yolume  V  of  his  Gesammelte  Werke. 
Gray:  Absolute  Measurements  in  Electricity  and   Magnetism.     A 

Treatise  on  Electricity  and  Magnetism. 


VI  SHORT    LIST    OF    WORKS. 

Green :  An  Essay  on  the  Application  of  Mathematical  Analysis  to 

the  Theories  of  Electricity  and  Magnetism. 

Harnack:  Grundlagen  der  Theorie  des  logarithmischen  Potentiales. 
Heaviside :  Electrical  Papers. 
Heine :  Kugelfunctionen. 

Helmholtz  :  Wissenschaftliche  Abhandlungen. 
Hertz  :   Gesammelte  Abhandlungen. 
Jordan  :  Cours  d' Analyse. 
Joubert,  Foster,  and  Atkinson :  Elementary  Treatise  on  Electricity  and 

Magnetism. 
Kirchhoff :   Gesammelte   Abhandlungen.     Yorlesungen    liber   mathe- 

matische  Physik. 

Elektricitat  und  Magnetismus.     Edited  by  Planck. 
Klein:  Vorlesungen  iiber  die  Potentialtheorie. 

Lame*  :  Legoiis  sur  les  Coordonnees  Curvilignes  et  leurs  Diverses  Appli 
cations. 
Mascart :   Traite"  d'Electricite"  Statique.     Also  Wallentin's  translation 

of  the  same  work  into  German,  with  additions. 
Mascart  et  Joubert :   Lemons  sur  1'Electricite"  et  le  Magnetisme.     Also 

Atkinson's  translation  of  the  same  work  into  English,  with 

additions. 
Mathieu  :   Theorie  du  Potential  et  ses  Applications  a  1'Electrostatique 

et  au  Magnetisme. 
Maxwell :  An   Elementary   Treatise  on  Electricity.     A  Treatise  on 

Electricity  and  Magnetism. 
Minchin  :  A  Treatise  on  Statics. 
Neumann,  C. :  Untersuchungen  iiber   das   logarithm! sche  und  New- 

ton'sche  Potential. 
Neumann,  F. :  Vorlesungen  tiber  die  Theorie  des  Potentials  und  der 

Kugelfunctionen. 

Nipher:  Electricity  and  Magnetism. 
Picard  :  Trait6  d' Analyse. 
Poincare* :  Electricite    et    Optique.       Les    Oscillations    filectriqueg. 

Theorie  du  Potential  Newtpnien. 

Riemann  :   Schwere,  Electricitilt  und  Magnetismus.     Edited  by  Hat- 
ten  dorff. 

Routh  :  Analytical  Statics. 
Schell :   Theorie  der  Bewegung  und  der  Kraf  te. 


SHORT    LIST    OF    WORKS.  Vll 

Steinmetz  :   Alternating  Current  Phenomena. 

Tarleton  :  The  Mathematical  Theory  of  Attraction. 

Thomson,  J.  J. :  Elements  of  Electricity   and  Magnetism.     Recent 

Researches  in  Electricity  and  Magnetism. 

Thomson,  W.  :   Reprint  of  Papers  on  Electrostatics  and  Magnetism. 
Thomson  and  Tait :  A  Treatise  on  Natural  Philosophy. 
Todhunter :  A  History  of  the  Mathematical  Theories  of  Attraction 

and   the   Figure   of   the    Earth.     The    Functions   of   Laplace, 

Bessel,  and  Lame. 

Turner  :  Examples  on  Heat  and  Electricity. 
Watson  and  Burbury :  The  Mathematical  Theory  of  Electricity  and 

Magnetism. 

Webster :  The  Theory  of  Electricity  and  Magnetism. 
Wiedemann  :  Die  Lehre  von  der  Elektricitat. 
Winkelmann  :  Handbuch  der  Physik. 


TABLE  OF  CONTENTS. 


CHAPTER   I. 

THE  ATTRACTION  OF  GRAVITATION. 
SECTION.  PAGE. 

1.  The  law  of  gravitation 1 

2.  The  attraction  at  a  point  .         .        .         ...        .         .1 

3.  The  unit  of  force .         .       2 

4.  The  attraction  due  to  discrete  particles     .        .        .        .        .  -    2 

5.  The  attraction,  at  a  point  in  its  axis,  due  to  a  straight  wire      .       3 

6.  The  attraction,  at  any  point,  due  to  a  straight  wire  ...      4 

7.  The  attraction,  at  a  point  in  its  axis,  due  to  a  cylinder  of 

revolution     .        .        .        .        .        •  •      •        •        .        .       7 

8.  The  attraction  at  the  vertex  of  a  cone  of  revolution,  due  to  the 

whole  cone  and  to  different  frusta        .        .        .        .        .8 

9.  The  attraction  due  to  a  homogeneous  spherical  shell ;    to  a 

solid  sphere •••        •        .11 

10.  The  attraction  due  to  a  homogeneous  hemisphere      .        .        .13 

11.  Apparent  anomalies  in  the  latitudes  of  places  near  the  foot  of  a 

hemispherical  hill 15 

12.  The  attraction  due  to  any  homogeneous  ellipsoidal  homoeoid  is 

zero  at  all  points  within  the  cavity  enclosed  by  the  shell      .     16 

13.  The  attraction  due  to  a  spherical  shell  the  density  of  which 

varies  with  the  distance  from  the  centre    .         .        .         .     18 

14.  The  attraction  at  any  point  due  to  any  given  mass     .        .         .19 

15.  The  component  in  any  direction  of  the  attraction  at  a  point  P, 

due  to  a  given  mass,  is  always  finite 21 

16.  The  attraction  between  two  straight  wires        .        .        .         .     22 

17.  The  attraction  between  two  spheres .23 

18.  The  attraction  between  any  two  rigid  bodies     .        ,        .         .24 
Examples . •        •        .25 

ix 


TABLE    OF    CONTENTS. 


CHAPTER    II. 

THE  NEWTONIAN   POTENTIAL   FUNCTION   IN   THE   CASE 
OF  GRAVITATION. 

SECTION.  PAGE. 

19.  Definition  of  the  Newtonian  potential  function          .         .         .20 

20.  The  derivatives  of  the  potential  function  relative  to  the  space 

coordinates  are  functions  of  these  coordinates,  which  repre 
sent  the  components,  parallel  to  the  coordinate  axes,  of  the 
attraction  at  the  point  (x,  y,  z) 30 

21.  Extension  of  the  statement  of  the  last  section  .         .         .         .31 

22.  The   potential    function   due   to   a  given    attracting   mass    is 

everywhere  finite,  and  the  statements  of  the  two  preced 
ing  sections  hold  good  for  points  within  the  attracting 
mass  ...........  32 

23.  The  potential  function  due  to  a  homogeneous  straight  wire       .     34 

24.  The  potential  function  due  to  a  homogeneous  spherical  shell     .     35 

25.  Equipotential  surfaces  and  their  properties       .         .         .         .37 

26.  The  potential  function  is  zero  at  infinity  .         .         .         .         .40 

27.  The  potential  function  as  a  measure  of  work     .         .         .         .41 

28.  Laplace's  Equation 44 

29.  The  second  derivatives  of  the  potential  function  are  finite  at 

points  within  the  attracting  mass 45 

30.  The  first  derivatives  of  the  potential  function  change  continu 

ously  as  the  point  (x,  y,  z)  moves  through  the  boundaries  of 
an  attracting  mass  .  .  .  .  .  .  .  .50 

31.  Theorem  due  to  Gauss 52 

32.  Tubes  of  force  and  their  properties    ......  55 

33.  Spherical  distributions  of  matter  and  their  attractions      .         .  56 

34.  Cylindrical  distributions  of  matter  and  their  attractions   .         .  60 

35.  Poisson's   Equation   obtained   by   the  application   of   Gauss's 

Theorem  to  volume  elements 61 

36.  Poisson's  Equation  in  the  integral  form          .         .         .         .66 

37.  The  average  value  of  the  potential  function  on  a  spherical 

surface.  The  potential  function  can  have  no  maxima  or 
minima  at  points  of  empty  space  ...  .  .  .67 

38.  The  equilibrium  of  fluids  at  rest  under  the  action  of  given 

forces   .         .         .-'«'.         .         .         .         .         .         .70 

Examples.         .         .         . 71 


TABLE    OF    CONTENTS.  XI 


CHAPTER   III. 

THE  NEWTONIAN   POTENTIAL  FUNCTION   IN   THE  CASE 
OF  REPULSION. 

SECTION.  PAGE. 

39.  Repulsion  according  to  the  Law  of  Nature       .         .         .         .75 

40.  The  force  at  any  point  due  to  a  given  distribution  of  repelling 

matter         .         .         .         .        .         .         .         .         .         .76 

41.  The  potential  function  due  to  repelling  matter,  as  a  measure 

of  work v       .        .        .78 

42.  Gauss's  Theorem  in  the  case  of  repelling  matter      .        .         .78 

43.  Poisson's  Equation  in  the  case  of  repelling  matter  ...       79 

44.  The  coexistence  of  two  kinds  of  active  matter         ...       80 
Examples         ...         ..         .         .        .        .        .         .82 


CHAPTER  IV. 

THE  PROPERTIES  OF  SURFACE  DISTRIBUTIONS.  GREEN'S 
THEOREM.  VECTORS.  THE  ATTRACTION  OF  ELLIPSOIDS. 
LOGARITHMIC  POTENTIAL  FUNCTIONS. 

45.  The  force  due  to  a  closed  shell  of  repelling  matter  ...       83 

46.  The  potential  function  is  finite  at  points  in  a  surface  distribu 

tion  of  matter     .         ...         .         ....        .85 

47.  The  normal  force  at  any  point  of  a  surface  distribution.     The 

pressure  on  the  surrounding  medium  .....       88 

48.  Green's   Theorem  and   its  applications.      Thomson's  Theo 

rem.  Dirichlet's  Principle.  Those  properties  of  the 
potential  function  which  are  sufficient  to  determine  the 
function  .  .  ; "  .  .91 

49.  Surface  distributions  which  are  equivalent  to  certain  volume 

distributions 109 

50.  Vectors.     Stokes's  Theorem.     The  derivatives  of  scalar  point 

functions Ill 

51.  The  attraction  due  to  homogeneous  ellipsoids          .         .         .     117 

52.  Logarithmic  potential  functions       .         .  ...         .     120 

Examples         .         .         .  .        .         .     *  .         .         .132 


TABLE    OF    CONTENTS. 


CHAPTER  V. 

THE  ELEMENTS  OP  THE  MATHEMATICAL  THEORY  OF 
ELECTRICITY. 

I.    ELECTROSTATICS. 
SECTION.  PAGE 

53.  Introductory ^5 

54.  The  charges  on  conductors  are  superficial        .         .  146 

55.  General  principles  which  follow  directly  from  the  theory  of 

the  Newtonian  potential  function      .        .        .  143 

56.  Tubes  of  force  and  their  properties .  150 

57.  Hollow  conductors  ......  152 

58.  The  charge  induced  on  a  conductor  which  is  put  to  earth  156 

59.  Coefficients  of  induction  and  capacity      .         .        .  157 

60.  The  distribution  of  electricity  on  a  spherical  conductor  .  .     159 

61.  The  distribution  of  a  given  charge  on  an  ellipsoidal  con 

ductor IQQ 

62.  Spherical  condensers .        .  161 

63.  Condensers  made  of  two  parallel  conducting  plates         .        .  164 

64.  The  capacity  of  a  long  cylinder  surrounded  by  a  concentric 

cylindrical  shell 166 

65.  The  charge  induced  on  a  sphere  by  a  charge  at  an  outside 

point 167 

66.  The  energy  of  charged  conductors 171 

67.  Composite  conductors .175 

68.  Specific  inductive  capacity.     Apparent  and  real  charges        .     176 

69.  Polarized  distributions.      Magnets.      Inductivity.      Suscepti 

bility.     Generalized  induction.     Polarized  shells.     Vector 
potential  functions  of  the  induction 185 

II.    ELECTROKINEMATICS. 

70.  Steady  currents  of  electricity 222 

71.  Linear  conductors.     Resistance.     Law  of  Tensions         .        .  226 

72.  Electromotive  force 230 

73.  Kirchhoff's  Laws.     The  Law  of  Divided  Circuits.     Wheat- 

stone's  net 234 

74.  The    heat  developed    in    a  circuit  which  carries  a  steady 

current  238 


TABLE    OF    CONTENTS. 

PAGE. 

SECTION. 

75.  Properties  of  the  potential  function  inside  conductors  which 

carry  steady  currents.     Heterogeneous  conductors    .         .241 

76.  Method  of  finding  cases  of  electrokinematic  equilibrium          .     246 


III.  ELECTROMAGNETISM. 

77.  The  electromagnetic  field  due  to  a  straight  current.     Straight 

currents  in  cylindrical  conductors       .....     251 

78.  Electromagnetic    fields    due    to    currents    in    closed    linear 

circuits 259 

79.  The  Law  of  Laplace.     The  mechanical  action  on  a  conductor 

which  carries  a  current  in  a  magnetic  field.     Electrokinetic 
energy '     ^62 

80.  The  electrodynamic  potential 

81.  Coefficients  of  self  and  mutual  induction         .         .         -         .278 

82.  Maxwell's  Current  Equations.      Solenoids.      Ring  Magnets. 

Hysteresis 281 

IV.  CURRENT  INDUCTION. 

83.  Electromagnetic  induction  and  its  laws 291 

84.  Superficial  induced  currents 2" 

85.  Variable  currents  in  single  circuits        ... 

86.  Alternate  currents  in  single  circuits 312 

87.  Variable    and    alternate    currents    in    neighboring    circuits. 

Transformers 

88.  The  general  equations  of  the  electromagnetic  field  .        .        .332 

337 

486 


337 
MISCELLANEOUS  PROBLEMS       


INDEX 


THE 


NEWTONIAN  POTENTIAL  FUNCTION. 

CHAPTER    I. 

THE  ATTRACTION  OP  GRAVITATION, 

1.  The  Law  of  Gravitation.      Every  body   in  the  universe 
attracts  every  other  body  with  a  force  which  depends  for  mag 
nitude  and  direction  upon  the  masses  of  the  two  bodies  and 
upon  their  relative  positions. 

An  approximate  value  of  the  attraction  between  any  two  rigid 
bodies  may  be  obtained  by  imagining  the  bodies  to  be  divided 
into  small  particles,  and  assuming  that  every  particle  of  the  one 
body  attracts  every  particle  of  the  other  with  a  force  directly 
proportional  to  the  product  of  the  masses  of  the  two  particles, 
and  inversely  proportional  to  the  square  of  the  distance  between 
their  centres  or  other  corresponding  points.  The  true  value  of 
the  attraction  is  the  limit  approached  by  this  approximate  value 
as  the  particles  into  which  the  bodies  are  supposed  to  be  divided 
are  made  smaller  and  smaller. 

2.  The  Attraction  at  a  Point.     By  ' '  the  attraction  at  any 
point  P  in  space,  due  to  one  or  more   attracting  masses,"  is 
meant  the  limit  which  would  be  approached  by  the  value  of  the 
attraction  on  a  sphere  of  unit  mass  centred  at  P  if  the  radius  of 
the  sphere  were  made  continually  smaller  and  smaller  while  its 
mass  remained  unchanged.     The  attraction  at  .P  is,  then,  the 
attraction  on  a  unit  mass  supposed  to  be  concentrated  at  P, 


2  THE    ATTRACTION    OF    GRAVITATION. 

If  the  attraction  at  every  point  throughout  a  certain  region 
has  a  value  other  than  zero,  the  region  is  called  "a  field  of 
force  "  ;  and  the  attraction  at  any  point  P  in  the  region  is  called 
"  the  strength  of  the  field  "  at  that  point. 

3.  The  Unit  of  Force.    It  will  presently  appear  that  all  spheres 
made  of  homogeneous  material  attract  bodies  outside  of  them 
selves  as  if  the  masses  of  the  spheres  were  concentrated  at  their 
middle  points.     If,  then,  k  be  the  force  of  attraction  between 
two  unit  masses   concentrated   at  points   at  the  unit  distance 
apart,  the  attraction  at  a  point  P  due  to  a  homogeneous  sphere 

of  radius  a  and  of  density  p  is  k  -  — r--A  where  r  is  the  dis- 

3  T 

tance  of  P  from  the  centre  of  the  sphere.  In  all  that  follows, 
however,  we  shall  take  as  our  unit  of  force  the  force  of  attrac 
tion  *  between  two  unit  masses  concentrated  at  points  at  the 
unit  distance  apart.  Using  these  units,  k  in  the  expression 
given  above  becomes  1,  and  the  attraction  between  two  particles 

of  mass  mi  and  w2  concentrated  at  points  r  units  apart  is     *2  2- 

4.  Attraction  due  to  Discrete  Particles.     The  attraction  at 
a  point  P,  due  to  particles  concentrated  at  different  points  in 

the  same  plane  with  P,  may  be  expressed 
in  terms  of  two  components  at  right 
angles  to  each  other. 

Let  the  straight  lines  joining  P  with 


\ 


— ~x 

the  different  particles  be  denoted  by  rl9 

rz>  ?*3?  •  •  •>  and  the  angles  which  these 
ms  lines   make   with   some   fixed   line  Px, 

FIG.  1.  by  ait  az>  ct3,  •  •  • .     If,  then,  the  masses 

*  These  are  called  "attraction  units  of  force."  When  the  attraction 
between  two  bodies  is  given  in  terms  of  absolute  kinetic  force  units  in  any 
system,  the  corresponding  value  of  k  is  sometimes  called  the  "  constant  of 
gravitation."  One  dyne  is  equivalent  to  about  1.543  X  107  c.g.s.  attrac 
tion  units  and  one  poundal  to  about  9.63  x  108  f.p.s.  attraction  units. 
For  simple  illustrative  problems  the  reader  may  consult  the  Miscella 
neous  Examples  at  the  end  of  the  book. 


THE   ATTRACTION   OF   GRAVITATION.  3 

of  the  several  particles  are  respectively  ??ix,  ra2,  ??i3,  •••,  the 
components  of  the  attraction  at  P  are 

Y-  _  ml  cos  ftj      ??i2  cos  a2  _  ^^  m  cos  a 


,  .?  ,  _a  rin 

~^T~  ~Z*~^~ 

in  the  direction  Px,  and 

^  _  W!  sin  Q!      Wgsinas  ,  _\^??i  sin  a  ro-, 

~          ~  ~~~ 


in  the  direction  Py,  perpendicular  to  Px. 
The  resultant  force  at  P  is 

P=VX2+F2,  [3] 

and  its  line  of  action  makes  with  Px  the  angle  whose  tangent 


If  the  particles  do  not  all  lie  in  the  same  plane  with  P,  we 
may  draw  through  P  three  mutually  perpendicular  axes,  and  call 
the  angles  which  the  lines  joining  P  with  the  different  particles 
make  with  the  first  axis  an  cu,  a3,  •••  ;  with  the  second  axis, 
ft?  ft?  ft?  •••  I  and  with  the  third  axis,  y,,  y2,  y3,  •••.  The  three 
components  in  the  directions  of  these  axes  of  the  attraction  at 
P  due  to  all  the  particles  are  then 

m  cos  a     v     \^m  cos/3  .    7  _X^W  cos 7        r/n 
1> —  5    *  —  /    ~> —  '   ^  —  7  3 — *       L^J 

ft  ~~^       /y^  ^^^       r§ 

The  resultant  attraction  is 


R  =  VX2+  F2+Z2,  [5] 

and  its  line  of  action  makes  with  the  axes  angles  whose  cosines 
are  respectively 

1 1' aQd  I'  M 

5.  Attraction  at  a  Point  in  the  Produced  Axis  of  a  Straight 
Wire.  Let  p.  be  the  mass  of  the  unit  of  length  of  a  uniform 
straight  wire  AB  of  length  /,  and  of  cross  section  so  small  that 


4  THE   ATTRACTION    OF    GRAVITATION. 

we  may  suppose  the  mass  of  the  wire  concentrated  in  its  axis 
(see  Fig.  2),  and  let  P  be  a  point  in  the  line  AB  produced  at  a 


M 
FIG.  2. 


distance  a  from  A.     Divide  the  wire  into  elements  of  length 
Ax.    The  attraction  at  P  due  to  one  of  these  elements,  M,  whose 

nearest  point  is  at  a  distance  x  from  P,  is  less  than  ^—  -  and 

x2 

greater  than  —  ^  -  - 
(8  +  A*)1 

The    attraction    at  P  due    to    the    whole   wire   lies  between 

^—  -  and    7   —  ^   X  ^  ;  but  these  quantities  approach  the 

x2  L^  (x  +  Ax)2 


same  limit  as  Ax  is  made  to  approach  zero,  so  that  the  attrac 
tion  at  P  is 


limit  X>AOJ_  C"      V-dx I1  I  1-71 


If  the  coordinates  of  P,  A,  and  5  are  respectively  (x,  0,  0), 
?!,  0,  0),  and  (xt  -f-  Z,  0,  0) ,  this  result  may  be  put  into  the  form 


r_i i     -i 

[_xl  —  x      xl  —  x  -f-  Ij 


6.  Attraction  at  any  Point,  due  to  a  Straight  Wire.  Let  P 
(Fig.  3)  be  any  point  in  the  perpendicular  drawn  to  the  straight 
wire  AB  at  A,  and  let  PA  =  c.  AB  =  I.  AM=  x.  and  the  angle 
ABP=  8.  Let  MN  be  one  of  the  equal  elements  of  mass  (/xAx) 
into  which  the  wire  is  divided,  and  call  PJtf,  r.  The  attraction 

at  P  due  to  this  element  is  approximately  equal  to  L^  and 

r- 

acts  in  some  direction  lying  between  PM  and  PN.    This  attrac 
tion  can  be  resolved  into  two  components  whose  approximate 

values  are     ^x'c     in  the  direction  PA,  and    ^x'x     in  the 


THE   ATTRACTION   OF   GRAVITATION. 


direction  PL.  The  true  values  of  the  components  in  these 
directions  of  the  attraction  at  P,  due  to  the  whole  wire,  are, 
then,  respectively  : 


and 


P] 


FIG.  3. 


The  resultant  attraction  is  equal  to  the  square  root  of  the  sura 
of  the  squares  of  these  components,  or 


and  its  line  of  action  makes  with  PA  an  angle  whose  tangent  is 
1  -  sin  8      1  -  cosAPB  2sin2±APB 


cos  8 


sin./LPB 


^  . 


That  is,  the  resultant  attraction  at  P  acts  in  the  direction  of 
the  bisector  of  the  angle  APB. 

From  these  results  we  can  easily  obtain  the  value  of  the 
attraction  at  any  point  P,  due  to  a  uniform  straight  wire  B'B 
(Fig.  4)  .  Drop  a  perpendicular  PA  from  P  upon  the  axis  of 
the  wire.  Let  AB  =  /,  AB'  =  V  ,  PA  =  c,  ABP=&,  AB'P  =  8', 
BPB'=6.  The  component  in  the  direction  PA  of  the  attrac 
tion  at  P  is  [9] 

ii 

-  (cosS  +  cos  8'), 


6  THE    ATTRACTION    OF    GRAVITATION, 

and  that  in  the  direction  PL  is 


so  that  the  resultant  attraction  is 


FIG.  4. 

The  line  of  action  PK  of  R  makes  with  PA  an  angle  <£  such 
that 

sin  Sf  —  sin  8 


and 


.'.B!PK=-- 


'-«)  =  £- 


It  is  to  be  noticed  that  PK  bisects  the  angle  0,  and  does  not 
in  general  pass  through  the  centre  of  gravity  or  any  other  fixed 
point  of  the  wire.  Indeed,  the  path  of  a  particle  moving  from 
rest  under  the  attraction  of  a  straight  wire  is  generally  curved  ; 
for  if  the  particle  should  start  at  a  point  Q  and  move  a  short 
distance  on  the  bisector  of  the  angle  BQB'  to  Q',  the  attraction 
of  the  wire  would  now  urge  the  particle  in  the  direction  of  the 
bisector  of  the  angle  BQ'B',  and  this  is  usually  not  coincident 
with  the  bisector  of  BQB'. 


THE   ATTRACTION    OF   GRAVITATION.  7 

If  q  is  the  area  of  the  cross  section  of  the  wire,  and  p  the 
mass  of  the  unit  volume  of  the  substance  of  which  the  wire  is 
made,  we  may  substitute  for  p  in  the  formulas  of  this  section 
its  value  qp. 

If  instead  of  a  very  thin  wire  we  had  a  body  in  the  shape  of 
a  prism  or  C3"linder  of  considerable  cross  section,  we  might 
divide  this  up  into  a  large  number  of  slender  prisms  and  use  the 
equations  just  obtained  to  find  the  limit  of  the  sum  of  the  attrac 
tions  at  an}'  point  due  to  all  these  elementary  prisms.  This 
would  be  the  attraction  due  to  the  given  body. 

7.  Attraction  at  a  Point  in  the  Produced  Axis  of  a  Cylinder 
of  Revolution.  In  order  to  find  the  attraction  due  to  a  homo 
geneous  cylinder  of  revolution  at  any  point  P  (Fig.  5)  in  the 
axis  of  the  cylinder  produced,  it  will  be  convenient  to  imagine 
the  cylinder  cut  up  into  discs  of  constant  thickness  Ac,  by 
means  of  planes  perpendicular  to  the  axis. 

Let  p  be  the  mass  of  the  unit  of  volume  of  the  cylinder,  and 
a  the  radius  of  its  base.  Consider  a  disc  whose  nearer  face  is 
at  a  distance  c  from  P,  and  divide  it  into  elements  by  means  of 

B'    B 


A     A 

FIG.  5. 

radial  planes  drawn  at  angular  intervals  of  A0  and  concentric 
cylindrical  surfaces  at  radial  intervals  of  Ar. 

The  mass  of  any  element  J/  whose  inner  radius  is  r  is  equal 
to  pAc-  A0[rA?*  4-  -J(Ar)2],  and  the  whole  attraction  at  P  due  to 

M  is  approximately  p  —  J  in  a  line  joining  P 

with  some  point  of  M.     The  component  of  this  attraction  in 
the  direction  PC  is  found  by  multiplying  the  expression  just 


8  THE    ATTRACTION    OF    GRAVITATION. 

^» 

given  by  -  ,  the  cosine  of  the  angle   CPS,  so  that  the 

Vc2  4-  1* 

attraction  at  P  in  the  direction  P<7,  due  to  the  whole  disc,  is 
approximately 


If  the  ]>ases  of  the  cylinder  are  at  distances  c0  and  c0  -f  7i 
from  P,  the  true  value  of  the  attraction  at  P  in  the  direction 
PC,  due  to  the  cylinder  QQ',  is 


dc 


Vc2-h  a2 
=  2  7rp[7i  +  Vc()2  +  a2  -  V(c0  +  /02+a2] .  [1 5] 


This  is  evidently  the  whole  attraction  at  P  due  to  the  cylin 
der,  for  considerations  of  symmetry  show  us  that  the  resultant 
attraction  at  P  has  no  component  perpendicular  to  PC. 

[14]  gives  the  attraction  due  to  the  elementary  disc  ABA'B', 
on  the  assumption  that  the  whole  matter  of  the  disc  is  concen 
trated  at  the  face  ABC.  The  actual  attraction  at  P  due  to 
this  disc  may  be  found  by  putting  c0  =  c  and  li  =  Ac  in  [15]. 

If  a,  the  radius  of  the  cylinder,  is  very  large  compared  with 
h  and  c0,  the  expression  [15]  for  the  attraction  at  P  due  to  the 
cylinder  approaches  the  value  2-Trph. 

8.  Attraction  at  the  Vertex  of  a  Cone.  The  attraction  due  to 
a  homogeneous  cone  of  revolution,  at  a  point  at  the  vertex  of 
the  cone,  may  be  found  by  the  aid  of  [14]. 

If  Fig.  6  represents  a  plane  section  of  the  cone  taken  through 
the  axis,  and  if  PM=  c,  MM'  =  Ac,  and  MB  =  r,  the  attraction 
at  P  due  to  the  disc  ABCD  is  approximately 

27rpAc     1 C  \=  27rpAc(l  —  COS  a), 

Vc2  +  r2J 


THE   ATTRACTION   OF    GRAVITATION. 


and  the  attraction  due  to  the  whole  cone  is 


2^(1  -  COSa)  AC  =  2irp(l  -  COSa) 

=  277/3(1  -COSa)  -PL.  [16] 

The  attraction  at  P  due  to  the  frustum  ABKN  is  found  by 
subtracting  the  value  of  the  attraction  due  to  the  cone  ABP 
from  the  expression  given  in  [16].  The  result  is 

2irp(l  -  cosa)  (PL  -P3/)  =  27rp(l-cosa)3/L,       [17] 

and  it  is  easy  to  see  from  this  that  discs  of  equal  thickness  cut 
out  of  a  cone  of  revolution  at  different  distances  from  the  vertex 
by  planes  perpendicular  to  the  axis  exert  equal  attractions  at 
the  vertex  of  the  cone. 


FIG.  6. 


It  follows  almost  directly  that  the  portions  cut  out  of  two 
concentric  spherical  shells  of  equal  uniform  density  and  equal 
thickness,  by  any  conical  surface  having  its  vertex  at  the 
common  centre  P  of  the  shells,  exert  equal  attraction  at  this 
centre  ;  but  we  may  prove  this  proposition  otherwise,  as  fol 
lows  : 

Divide  the  inner  surface  of  the  portion  cut  out  of  one  of  the 
shells  by  the  given  cone  into  elements,  and  make  the  perimeter 
of  each  of  these  surface  elements  the  directrix  of  a  conical 
surface  having  its  vertex  at  P.  Divide  the  given  shells  into 
elementary  shells  of  thickness  Ar  by  means  of  concentric  spheri 
cal  surfaces  drawn  about  P.  In  this  way  the  attracting  masses 
will  be  cut  up  into  volume  elements. 

Let  ML'  (Fig.  7)  represent  one  of  these  elements,  whose 
inner  surface  has  a  radius  equal  to  r ;  then,  if  the  elementary 


10  THE    ATTRACTION    OF    GRAVITATION. 

cone  APB  intercept  an  element  of  area  Aw  from  a  spherical  sur 
face  of  radius  unity  drawn  around  P,  the  area  of  the  surface 
element  at  MM'  is  ?~Au>,  and  that  at  LL'  is  (r  +  A?*)2 Aw.  The 


attraction  at  P  in  the  direction  PM,  due  to  the  element  ML1,  is 
approximately 

p 


and  the  component  of  this  in  any  direction  Px,  making  an 
angle  a  with  PJf,  is  approximately  p  Aw  A?'  cos  a.  The  attraction 
at  P  in  the  direction  Px,  due  to  the  whole  shell  EDFG,  is, 
then,  ^-N 

X  =  lim        p  A?'  Aw  cos  a, 


where  the  sum  is  to  include  all  the  volume  elements  which  go  to 
make  up  the  shell.  If  PF=r0,  PG  =  i\,  PP'=r0',  PG'  =  r1', 
and  jL  =  FG  = 


X=  I  pdr  I  cosadw  =  p/A  I 


The  attraction  at  P  in  the  same  direction,  due  to  the  shell 
E'D'F'G',  is 

X'  =  p  I    l  dr  I  cosadw  =  pp  f  cos  adw. 

But  the  limits  of  integration  with  regard  to  <o  are  the  same  in 
both  cases  ;  .-.  X=  X',  which  was  to  be  proved. 

If  the  shells  are  of  different  thicknesses,  it  is  evident  that 
they  will  exert  attractions  at  P  proportional  to  these  thick 
nesses. 


THE   ATTRACTION    OF    GRAVITATION. 


11 


The  area  of  the  portion  which  a  conical  surface  cuts  out  of  a 
spherical  surface  of  unit  radius  drawn  about  the  vertex  of  the 
cone  is  called  u  the  solid  angle  "  of  the  conical  surface. 

9.  Attraction  of  a  Spherical  Shell.  In  order  to  find  the 
attraction  at  P,  any  point  in  space,  due  to  a  homogeneous 
spherical  shell  of  radii  r0  and  ?*1?  it  will  be  best  to  begin  by 
dividing  up  the  shell  into  a  large  number  of  concentric  shells 
of  thickness  Ar,  and  to  consider  first  the  attraction  of  one  of 
these  thin  shells,  whose  inside  radius  shall  be  r. 

Let  p  be  the  density  of  the  given  shell,  that  is,  the  mass  of 
the  unit  of  volume  of  the  material  of  which  the  shell  is  com 
posed.  Join  P  (Fig.  8)  with  0  by  a  straight  line  cutting  the 
inner  surface  of  the  thin  shell  at  JV,  and  pass  a  plane  through 
PO  cutting  this  inner  surface  in  a  great  circle  NLSL',  which 


FIG.  8. 

will  serve  as  a  prime  meridian.  Using  N  as  a  pole,  describe 
upon  the  inner  surface  of  the  thin  shell  a  number  of  parallels  of 
latitude  so  as  to  cut  off  equal  arcs  on  NLSL'.  Denote  by  A0 
the  angle  which  each  one  of  these  arcs  subtends  at  0.  Through 
PO  pass  a  numbei  of  planes  so  as  to  cut  up  each  parallel  of 
latitude  into  equal  arcs.  Denote  by  A<£  the  angle  between  any 
two  contiguous  planes  of  this  series.  By  this  means  the  inner 
surface  of  the  elementary  shell  will  be  divided  into  small  quad 
rilaterals,  each  of  which  will  have  two  sides  formed  of  meridian 
arcs,  of  length  r-A4,  and  two  sides  formed  of  arcs  of  parallels 
of  latitude,  of  length  rsin0-A<£  and  /-sin((9  -f  A0)«  A</>,  where 


12  THE   ATTRACTION    OF    GRAVITATION. 

9  is  the  angle  which  the  radius  drawn  to  the  parallel  of  higher 
latitude  makes  with  ON.  The  area  of  one  of  these  quadri 
laterals  is  approximately  rs'mO-  A0-  A<£,  and  the  thickness  of 
the  shell  is  Ar,  so  that  the  element  of  volume  is  approxi 
mately  r2sin#-  Ar-  A(9  •  A^>.  Let  PM=i/,  then  the  attrac 
tion  at  P,  due  to  an  element  of  mass  which  has  a  corner  at 


A  i     pr2  sin  6  Ar  >     .      ,,       ,. 

M,   is   approximately  '  —  —      --,   in   the  direction  PM. 

This  force  ma}*  be  resolved  into  three  components  :  one  in  the 
direction  PO,  the  others  in  directions  perpendicular  to  PO 
and  to  each  other  ;  but  it  is  evident  from  considerations  of 
symmetry  that  in  finding  the  attraction  at  P  due  to  the  whole 
shell  we  shall  need  only  that  component  which  acts  in  PO.  This 


.  ,  '  -  cosKPM          .,,  7^ 

is  approximately  '-  —  —-  —  ;  or,  if  PO  =  c, 

*J 

Pr2sin0(c  — 


f 

The  attraction  at  P  due  to  the  whole  elementary  shell  is,  then, 
approximately  (truly  on  the  assumption  that  the  whole  mass  of 
the  shell  is  concentrated  at  its  inner  surface)  , 

Ar  f  rpr2smO(c-rc 

J  J  if 

and  the  true  value  at  P  of  the  attraction  due  to  the  given  shell  is 


C^ 
Jr 


Xdr.  [20] 

If  in  the  expression  for  X  we  substitute  for  0  its  value  in 
terms  of  2/,  we  have,  since 

if  =  (?  -f-  r2  —  2  GTCOS0, 
and  hence  2ydy  =  2cr  sin  0  dO, 


THE   ATTRACTION    OF   GRAVITATION.  13 

In  order  to  find  the  limits  of  the  integration  with  regard  to  y, 
we  must  distinguish  between  two  cases  : 

I.  If  P  is  a  point  in  the  cavity  enclosed  by  the  given  shell, 

y0  =  r  —  c     and     y-^  =  r  +  c  ; 

~~  ^  +  0'  +  c)2      ?~  ~  G~  +  (*'  ~  °)n  =  Q,     [22] 
r  +  c  r  —  c 

friJXar==0;  [23] 

«^r0 

so  that  a  homogeneous  spherical  shell  exerts  no  attraction  at 
points  in  the  cavity  which  it  encloses. 

II.  If  P  is  a  point  without  the  given  shell, 

yQ=c  —  r     and     yl  =  c  +  r  ; 
2  +  (c+r)2      r2—  c2+(c  —  r)  - 


and 


c  H-  r  c  —  r 

and  friXdr  =  -  ^  (r^-  ?-03) .  [25] 

i  O^^1"'  LJ 


From  this  it  follows  that  the  attraction  due  to  a  spherical 
shell  of  uniform  density  is  the  same,  at  a  point  without  the  shell, 
as  the  attraction  due  to  a  mass  equal  to  that  of  the  shell  con 
centrated  at  the  shell's  centre. 

If  in  [25]  we  make  r0  =  0,  we  have  the  attraction,  due  to  a 
solid  sphere  of  radius  i\  and  density  p.  at  a  point  outside  the 
sphere  at  a  distance  c  from  the  centre.  This  is 


10.  Attraction  due  to  a  Hemisphere.  At  any  point  P  in  the 
plane  of  the  base  of  a  homogeneous  hemisphere,  the  attraction 
of  the  hemisphere  gives  rise  to  two  components,  one  directed 
toward  the  centre  of  the  base,  the  other  perpendicular  to  the 
plane  of  the  base.  We  will  compute  the  values  of  these  com 
ponents  for  the  particular  case  where  P  lies  on  the  rim  of  the 
hemisphere's  base,  and  for  this  purpose  we  will  take  the  origin 


14 


THE    ATTRACTION    OF    GRAVITATION. 


of  our  s}'stem  of  polar  coordinates  at  P.  because  by  so  doing 
we  shall  escape  having  to  deal  with  a  quantity  which  becomes 
infinite  at  one  of  the  limits  of  integration.  Denote  the  coordi 
nates  of  any  point  L  in  the  hemisphere  by  r,  0,  </>,  where  (Fig.  9) 
XPN=  </>,  IPL  =  0,  and  PL  =  r. 


FIG.  9. 

If  TI  be  the  radius  of  the  hemisphere, 
PT=  PNcosNPT  =  PXcos  XPN-  co 


=  2i\  sin0  cos<£. 


cos 


.    »  •     , 
=  snip  sin  <£. 


The  mass  of  a  polar  element  of  volume  whose  corner  is  at 
L  is  approximately  p-  IL\<j>-  PLM-  A?-  or  p^sin^ArA^A^, 
and  this  divided  by  r2  is  the  attraction  at  P  in  the  direction  PL 
of  the  element,  supposed  concentrated  at  L.  The  components 
of  this  attraction  in  the  direction  PX  and  PFare  respectively 
/3sin0ArA0A<£cosXP.L  and  p  siii0Ar  A0A0COS/SP.L. 

The  component  in  the  direction  Py  of  the  attraction  at  P  due 
to  the  whole  hemisphere  is,  then, 


—          f*n        ,<»    7'j  sn 

2d</>  I  d6  \  p  sin20 


sin  0  cos  4> 


[27] 


THE   ATTRACTION    OF    GRAVITATION.  15 

and  the  component  in  the  direction  Px  is 

J—  (**       /»2  Tj  sin  0  cos  <f> 

2 d(j>  (  dO  \  psm20cos<j>dr  =  f  w/>?v  [28] 

This  last  expression  might  have  been  obtained  from  [26]  by 
making  c  equal  to  r  and  halving  the  result. 

11.  Attraction  of  a  Hemispherical  Hill.  If  at  a  point  on  the 
earth  at  the  southern  extremity  of  a  homogeneous  hemispheri 
cal  hill  of  densit}*  p  and  radius  i\  the  force  of  gravity  due  to  the 
earth,  supposed  spherical,  is  g,  the  attraction  due  to  the  earth 
and  the  hill  will  give  rise  to  two  components,  g  —  ^pi\  down 
wards,  and  f  Trpi\  northwards.  The  resultant  attraction  does 
not  therefore  act  in  the  direction  of  the  centre  of  the  earth,  but 

o 

makes  with  this  direction  an  angle  whose  tangent  is 


FIG.  10. 


Let  <£  (Fig.  10)  be  the  true  latitude  of  the  place  and  (<£  —  a) 
the  apparent  latitude,  as  obtained  by  measuring  the  angle  which 
the  plumb-line  at  the  place  makes  with  the  plane  of  the  equator. 
Let  a  be  the  radius  of  the  earth  and  o-  its  average  density.  Then 


tena= 


[29] 


16 


THE    ATTRACTION    OF   GRAVITATION. 


The   radius   of  the  earth  is   very  large  compared   with    the 
radius  of  the  hill,  and  a  is  a  small  angle,  so  that  approximately 


a  =  --,  and  the  apparent  latitude  of  the  place  is  <£ 

2  a<r  2  ao- 

If  fa  is  the  true  latitude  of  a  place  just  north  of  the  same  hill, 
its  apparent  latitude  will  be  fa  +  -^-^  ,  and  the  apparent  differ- 


ence  of  latitude  between  the  two  places,  one  just  north  of  the 
hill  and  the  other  just  south  of  it,  will  be  the  true  difference 

plus  £-!.     If  there  were  a  hemispherical  cavity  between  the  two 

dor 

places  instead  of  a  hemispherical  hill,  the  apparent  difference  of 
latitude  would  be  less  than  the  true  difference. 

12.  Ellipsoidal  Homceoids.  A  shell,  thick  or  thin,  bounded 
by  two  ellipsoidal  surfaces,  concentric,  similar,  and  similarly 
placed,  shall  be  called  an  ellipsoidal  homosoid. 

It  is  a  property 
of  every  such 
shell  that  if  any 
straight  line  cut 
its  outer  surface 
at  the  points  S,  S' 
(Fig.  11)  and  its 
inner  surface  at 
Q,  Q',  so  that  these 
four  points  lie  in 
the  order  SQQ'S', 
the  length  SQ  will 
FIG.  11.  be  equal  to  the 

length   Q'S'* 
We  will  prove  that  the  attraction  of  a  homogeneous  closed 


*  The  section  of  the  homceoid  made  by  a  plane  which  passes  through 
the  centre  and  the  secant  line,  is  bounded  by  two  concentric,  similar,  and 
similarly  placed  ellipses.  This  figure  may  be  regarded  as  an  orthogonal 
projection  of  two  concentric  circles  cut  by  a  straight  line. 


THE    ATTRACTION   OF   GRAVITATION.  17 

ellipsoidal  hoimroid,  at  any  point  P  in  the  cavity  which  it  shuts 
in,  is  zero. 

Make  P  the  vertex  of  a  slender  conical  surface  of  two 
nappes,  A  and  B,  and  suppose  the  plane  of  the  paper  to  be 
so  chosen  that  PQ  is  the  shortest  and  PM  the  longest  length 
cut  from  any  element  of  the  nappe  A  by  the  inner  surface  of 
the  homoeoid.  Draw  about  P  spherical  surfaces  of  radii  PQ. 
PM,  PS,  and  PO,  and  imagine  the  space  between  the  inner 
most  and  outermost  of  these  surfaces  filled  with  matter  of  the 
same  density  as  the  homoeoid.  The  nappe  A  cuts  out  a  portion 
from  this  spherical  shell  whose  trace*  on  the  plane  of  the 
paper  is  QLOT.  Let  us  call  this,  for  short,  "the  element 
QLOT."  The  attraction  at  P,  due  to  the  element  QMOS  which 
A  cuts  out  of  the  homoeoid,  is  less  than  the  attraction  at  the 
same  point  due  to  the  element  QLOT,  and  greater  than  that 
due  to  the  element  whose  trace  is  KMNS.  But  the  attraction 
at  P,  due  to  the  first  of  these  elements  of  spherical  shells,  is  to 
the  attraction  due  to  the  other  as  the  thickness  of  the  first  shell 
is  to  that  of  the  other,  or  as  Q77is  to  KS.  (See  Section  8.) 
The  limit  of  the  ratio  of  QT  to  KS,  as  the  solid  angle  of  the 
cone  is  made  smaller  and  smaller,  is  unity ;  therefore  the  limit 
of  the  ratio  of  the  attraction  at  P  due  to  the  element  QMOS,  to 
the  attraction  due  to  the  element  of  spherical  shell  whose  trace 
is  QLNS,  is  unity.  By  a  similar  construction  it  is  easy  to  show 
that  the  limit  of  the  ratio  of  the  attraction  at  P,  due  to  the 
element  which  B  cuts  out  of  the  homoeoid,  to  the  attraction  due 
to  the  portion  of  spherical  shell  whose  trace  is  Q'L'N'S',  is 
unity. 

But  the  attractions  at  P,  due  to  the  elements  Q'L'N'S'  and 
QLNS,  are  equal  in  amount  (since  their  thicknesses  are  the 
same)  and  opposite  in  direction,  so  that  if  for  the  elements  of 
the  homoeoid  these  elements  were  substituted,  there  would  be  no 
resultant  attraction  at  P.  In  order  to  get  the  attraction  at  P 
in  any  direction  due  to  the  whole  homoeoid  we  may  cut  up  the 
inner  surface  of  the  homoeoid  into  elements,  use  the  perimeter 
of  each  one  of  these  elements  as  the  directrix  of  a  conical  sur- 


18  THE   ATTRACTION    OF    GRAVITATION. 

face  having  its  vertex  at  P,  and  find  the  limit  of  the  sum  of  the 
attractions  due  to  the  elements  which  these  conical  surfaces  cut 
from  the  homoaoid.  Wherever  we  have  to  find  the  finite  limit  of 
the  sum  of  a  series  of  infinitesimal  quantities,  we  may  without 
error  substitute  for  any  one  of  these  another  infinitesimal,  the 
limit  of  whose  ratio  to  the  first  is  unity.  For  the  attractions  at  P 
due  to  the  elements  of  the  homoeoid  we  may,  therefore,  substi 
tute  attractions  due  to  elements  of  spherical  shells,  which,  as  we 
have  seen,  destroy  each  other  in  pairs.  Hence  our  proposition. 
A  shell  bounded  by  two  concentric  spherical  surfaces  gives  a 
special  case  under  this  theorem. 

13.  Sphere  of  Variable  Density,  The  density  of  a  homo 
geneous  body  is  the  amount  of  matter  contained  in  the  unit 
volume  of  the  material  of  which  the  body  is  composed,  and  this 
may  be  obtained  by  dividing  the  mass  of  the  body  by  its  volume. 

If  the  amount  of  matter  contained  in  a  given  volume  is  not 
the  same  throughout  a  body,  the  body  is  called  heterogeneous, 
and  its  density  is  said  to  be  variable. 

The  average  density  of  a  heterogeneous  body  is  the  ratio  of 
the  mass  of  the  body  to  its  volume.  The  actual  density  p  at 
any  point  Q  inside  the  body  is  defined  to  be  the  limit  of  the 
ratio  of  the  mass  of  a  small  portion  of  the  body  taken  about  Q 
to  the  volume  of  this  portion  as  the  latter  is  made  smaller  and 
smaller. 

The  attraction,  at  any  point  P,  due  to  a  spherical  shell  whose 
density  is  the  same  at  all  points  equidistant  from  the  common 
centre  of  the  spherical  surfaces  which  bound  the  shell  but  dif 
ferent  at  different  distances  from  this  centre,  may  be  obtained 
with  the  help  of  some  of  the  equations  in  Article  9. 

Since  p  is  independent  of  0  and  <£,  it  may  be  taken  out  from 
under  the  signs  of  integration  with  regard  to  these  variables, 
although  it  must  be  left  under  the  sign  of  integration  with  re 
gard  to  r. 

Equations  19  to  24  inclusive  hold  for  the  case  that  we 
are  now  considering  as  well  as  for  the  case  when  p  is  constant, 


THE   ATTRACTION   OF    GRAVITATION. 


19 


so  that  the  attraction  at  all  points  within  the  cavity  enclosed  by 
a  spherical  shell  whose  density  varies  with  the  distance  from  the 
centre  is  zero. 

If  P  is  without  the  shell,  the  attraction  is 


^•1/(rVr«dr. 


or,  if  P=/(r), 


The  mass  of  the  shell  is  evidently 
limit 


^  ViVr2  -f(r)dr  =  ±TT  Cf(r) 

'  U^^r  Jr 


[30] 


[31] 


and  [30]  declares  that  a  spherical  shell  whose  density  is  a 
function  of  the  distance  from  its  centre  attracts  at  all  outside 
points  as  if  the  whole  mass  of  the  shell  were  concentrated  at  the 
centre. 

If  rQ  =  0,  we  have  the  case  of  a  solid  sphere. 

14.  Attraction  due  to  any  Mass.  In  order  to  find  the  attrac 
tion  at  a  point  P  (Fig.  12),  due  to  any  attracting  masses  Jf,  we 
may  choose  a  system  of  rectangular  coordinate  axes  and  divide 


FIG.  12. 

M'  up  into  volume  elements.  If  p  is  the  average  density  of  one 
of  these  elements  (Ar'),  the  mass  of  the  element  will  be  pAv'. 
Let  Q,  whose  coordinates  are  a?',  y',  z',  be  a  point  of  the  ele- 


20  THE    ATTRACTION    OF   GRAVITATION. 

mcnt,  and  let  the  coordinates  of  P  be  x,  ?/,  z.     The  attraction 
at  P  in  the  direction  PQ  due  to  this  element  is  approximately 


—  2,  and  the  components  of  this  in  the  direction  of  the  coordi 
nate  axes  are 


-cos/8',     and  ^-  cos/,  [32] 

PQ"  PQ~  PQ 

where  a',  /3',  yr  are  the  angles  which  PQ  makes  with  the  positive 
directions  of  the  axes. 
It  is  easy  to  see  that 

PL      x'-x 


and,  similarly,  that 


y<  y 

cos  ft'  =  " -J  ?      and     cos  y '  = 


' 


PQ  PQ 

Moreover, 


and  this  we  will  call  r2. 

The  true  values  of  the  components  in  the  direction  of  the 
coordinate  axes  of  the  attraction  nt  P,  due  to  all  the  elements 
which  go  to  make  up  M'  ,  are,  then, 

/(V—  a;) 


P(x'-x)dx'dy'dz' 


C  C  C  P 

J  J  J  [(«'-  x-) 


limit        p&v'(y'—  y} 


=  f  f  f  P(y'-y}dx'dy'dz'  f33  -, 

J  J  J  [(x'-a;)2+(y-2/)2+(^-^)2]^ 
_    limit  V^pA?/(2'—  z) 


r  r  r      P(z'-z)dx'dy'dz'      .    r33  ., 

J  J  J  [(«'-  xy+  (y'-  yy+  (z>-  zyy*  ' 


THE   ATTRACTION    OF    GRAVITATION.  21 

where  p  is  the  density  at  the  point  (V,  //',  z'),  and  where  the 
integrations  with  regard  to  a;', .?/,  and  z'  are  to  include  the  whole 
of  M'. 

The  resultant  attraction  at  P,  due  to  J/%  is 


fF2  +  Z2;  [34] 

and  its  line  of  action  makes  with  the  coordinate  axes  angles 
whose  cosines  are 


The  component  of  the  attraction  at  the  point  (x,  y,  z)  in  a 
direction  making  an  angle  e  with  the  line  of  action  of  R  is 
R cose.  If  the  direction  cosines  of  this  direction  are  A',  /u',  v', 
we  have 

COS  e  =  AA'  -f  /Jifji'  -f-  vv'. 

15.  The  quantities  X,  F,  Z,  and  7?,  which  occur  in  the  last 
section,  are  in  general  functions  of  the  coordinates  &,  i/,  and  z  of 
the  point  P.  Let  us  consider  X,  whose  value  is  given  in  [33A] . 

x'  -  x 


If  P  lies  without  the  attracting  mass  3/',  the  quantity   - 

is  finite  for  all  the  elements  into  which  3/'  is  divided.  Let  L 
be  the  largest  value  which  it  can  have  for  an}*  one  of  these 

elements,  then  X  is  less  than  L  \    \    \  pdx'dy'dz',  or  L-M1,  and 

this  is  finite.  If  P  is  a  point  within  the  space  which  the  attract 
ing  mass  occupies,  it  is  easy  to  show  that,  whatever  physical 
meaning  we  may  attach  to  X,  it  has  a  finite  value.  To  prove 
this,  make  P  the  origin  of  a  system  of  polar  coordinates,  and 
divide  M'  up  into  elements  like  those  used  in  Section  10.  It 
will  then  be  clear  that 

X=  (  (  CPaw*0co8<t>drd8d<t>,  [3(5] 

where  the  limits  are  to  be  chosen  so  as  to  include  all  the  at 
tracting  mass.  Since  sin:>^cos</)  can  never  be  greater  than 


22  THE    ATTRACTION   OF   GRAVITATION. 

unity,  X  is  less  than   J   J   jp(2r<20cty,  which  is  evidently  finite 

when  p  is  finite,  as  it  always  is  in  fact. 
The  corresponding  expressions, 

Y=  C  C  CPsiu2Osin<l>drdOd<f>,  [37] 

and  Z  =  C  f  CP  sin  0  cos  Odrd6d<j>,  [38] 

can  be  proved  finite  in  a  similar  manner ;  and  it  follows  that 
X,  y,  Z,  and  consequently  R,  are  finite  for  all  values  of  cc,  ?/, 
and  z. 

As  a  special  case,  the  attraction  at  a  point  P  within  the  mass 
of  a  homogeneous  spherical  shell,  of  radii  r0  and  r1?  and  of  den 
sity  p,  is 

f«£         ™  3\ 

[39] 


where  r  is  the  distance  of  P  from  the  centre  of  the  shell. 

16.  Attraction  between  Two  Straight  Wires.  Let  AK  and 
B  K'  (Fig.  13)  be  two  straight  wires  of  lengths  I  and  V  and  of 
line-densities  x  and  !  ;  and  let  KB  =  c.  Divide  AK  into 


K' 


M  M' 

FIG.  13. 

elements  of  length  A#,  and  consider  one  of  these  MM'  ,  such 
that  AM  =x.     The  attraction  of  BK1  on  a  unit  mass  concen 


trated  at  M  would  be  (Sections  2  and  5),  /*'  --       [•     If, 

therefore,  the  whole  element  MM'  whose  mass  is  /xA#  were  con 
centrated  at  3f,  the  attraction  on  it,  due  to  BK',  would  be 


THE    ATTRACTION    OF    GRAVITATION. 


The  actual  force,  due  to  the  attraction  of  BK'  ,  with  which  the 
whole  wire  AK  is  urged  toward  the  right,  is 


Jo    (p-(l  +  l'+c)  ~  X-(1+C)J 


17.  Attraction  between  Two  Spheres.  Consider  two  homo 
geneous  spheres  of  masses  M  and  J/'  (Fig.  14),  whose  centres 
C  and  C"  are  at  a  distance  c  from  each  other.  Divide  the  sphere 
M'  into  elements  in  the  manner  described  in  Section  9.  The 
attraction  due  to  3/at  any  point  P'  outside  of  this  sphere  is,  as 


we  have  seen 
PC. 


and  its  line  of  action  is  in  the  direction 


FIG.  14. 

Let  P'=(r,  0,0)  be  any  point  in  the  sphere  3ff,  and  let 
CP'  =  y.  The  attraction  of  M  in  the  direction  PC  on  an 
element  of  mass  p^^sin^A/*  A0A0  supposed  concentrated  at  P  is 


and  the  component  of  this  parallel  to  the 


line  VC  is 


for<je 


24  THE   ATTRACTION    OF    GRAVITATION. 

which  the  whole  sphere  M1  is  urged  toward  the  right  by  the 
attraction  of  M  is,  then, 

^rcos0); 


where   the   integration   is   to  be  extended   to   all  the  elements 
which  go  to  make  up  M'.     It  is  proved  in  Section  9  that  the 

M' 

value  of  this  triple  integral  is  —5-,  so  that  the  force  of  attraction 

C" 

.    MM' 

between  the  two  spheres  is  — - — 


18.  Attraction  between  any  Two  Rigid  Bodies.  In  order  to 
find  the  force  with  which  a  rigid  body  M  is  pulled  in  any  direc 
tion  (as  for  instance  in  that  of  the  axis  of  x)  by  the  attraction 
of  another  body  M',  we  must  in  general  find  the  value  of  a 
sextuple  integral. 

Let  M  be  divided  up  into  small  portions,  and  let  Aw  be  the 
mass  of  one  of  these  elements  which  contains  the  point (#,?/, z). 

The  component  in  the  direction  of  the  axis  of  x  of  the  attrac 
tion  at  (x,  ?/,  z)  due  to  M'  is 

p(x'-x)dx'dy'dz' 


and  this  would  be  the  actual  attraction  in  this  direction  on  a 
unit  mass  supposed  concentrated  at  (x,  y,  z}.  If  the  mass  Am 
were  concentrated  at  this  point,  the  attraction  on  it  in  the  direc 
tion  of  the  axis  of  x  would  be 

Am  f  f  f  _  p(x'-x)dx'dy'M_  [43] 

JJJ  [(X'-x)*+(y'-y)*+(z'-z)*]t 


The  actual  attraction  in  the  direction  of  the  axis  of  x  of  M  ' 
upon  the  whole  of  M  is,  then, 

liniit  p(*'-x)dx'dy'tl*' 


THE   ATTRACTION   OF   GRAVITATION.  25 

If  p1  is  the  density  at  the  point  (x,  y,  z) ,  and  if  the  elements 
into  which  M  is  divided  are  rectangular  parallelepipeds  of  di 
mensions  A#,  A#,  and  Az,  the  expression  just  given  may  be 
written 

P'P(X>-  x)dxdydzdx'dyW  (-45-1 

'*' 


where  the  integrations  are   first  to  be   extended  over  J/'  and 
then  over  J/. 

EXAMPLES. 

1.  Find  the  resultant  attraction,  at  the  origin  of  a  system  of 
rectangular  coordinates,  due  to  masses  of  12,  16,  and  20  units 
respectively,  concentrated  at  the  points  (3,  4),  (— 5,  12),  and 
(8,  —6).     What  is  its  line  of  action  ? 

2.  Find  the  value,  at  the  origin  of  a  system  of  rectangular 
coordinates,  of  the  attraction  due  to  three  equal  spheres,  each  of 
mass  m,   whose  centres   are  at  the  points  (a,  0,  0),  (0,  &,  0), 
(0,  0,  c) .     Find  also  the  direction-cosines  of  the  line  of  action 
of  this  resultant  attraction. 

3.  Show  that  the  attraction,  due  to  a  uniform  wire  bent  into 
the  form  of  the  arc  of  a  circumference,  is  the  same  at  the  centre 
of    the   circumference    as   the    attraction  due    to    any   uniform 
straight  wire  of  the  same  density  which  is  tangent  to  the  given 
wire,  and  is  terminated  by  the  bounding  radii  (when  produced) 
of  the  given  wire. 

4.  Show  that  in  the  case  of  an  oblique  cone  whose  base  is 
any  plane  figure  the  attraction  at  the  vertex  of  the  cone  due  to 
any  frustum  varies,  other  things  being  equal,  as  the  thickness 
of  the  frustum. 

5.  Find  the  equation  of  a  family  of  surfaces  over  each  one  of 
which  the  resultant  force  of  attraction  due  to  a  uniform  straight 
wire  is  constant. 

6.  Using  the  foot-pound-second  system  of  fundamental  units, 
and  assuming  that  the  average  density  of  the  earth  is  5.6,  com 
pare  with  the  poundal  the  unit  of  force  used  in  this  chapter. 


26  THE   ATTRACTION    OF   GRAVITATION. 

7.  If  in  Fig.  2  we  suppose  P  moved  up  to  A,  the  attraction 
at  P  becomes  infinite  according  to  [7],    and  yet  Section  15 
asserts  that  the  value,  at  any  point  inside  a  given  mass,  of  the 
attraction  due  to  this  mass  is  always  finite.     Explain  this. 

8.  A  spherical  cavity  whose  radius  is  r  is  made  in  a  uniform 
sphere  of  radius  2  r  and  mass  m  in  such  a  way  that  the  centre 
of  the  sphere  lies  on  the  wall  of  the  cavity.    Find  the  attraction 
due  to  the  resulting  solid  at  different  points  on  the  line  joining 
the  centre  of  the  sphere  with  the  centre  of  the  cavity. 

9.  A  uniform  sphere  of  mass  m  is  divided  into  halves  by  the 
plane  AB  passed  through  its  centre  C.     Find  the  value  of  the 
attraction  due  to  each  of  these  hemispheres  at  P,  a  point  on  the 
perpendicular  erected  to  AB  at  (7,  if  CP  =  a. 

10.  Considering  the  earth  a  sphere  whose  density  varies  only 
with  the  distance  from  the  centre,  what  may  we  infer  about  the 
law  of  change  of  this  density  if  a  pendulum  swing  with  the  same 
period  on  the  surface  of  the  earth  and  at  the  bottom  of  a  deep 
mine  ?     What  if  the  force  of  attraction  increases  with  the  depth 

at  the  rate  of  -th  of  a  dyne  per  centimetre  of  descent? 
n 

11.  The  attraction  due  to  a  cylindrical  tube  of  length  h  and 
of  radii  EQ  and  R±,  at  a  point  in  the  axis,  at  a  distance  c0  from 
the  plane  of  the  nearer  end,  is 

27rp[Vc02  +  ^12-V^+^2+V(c0  +  /02  +  ^o2-V(c0  +  /02  +  ^12]. 

[Stone.] 

12.  A  spherical  cavity  of  radius  b  is  hollowed  out  in  a  sphere 
of   radius   a   and   density  p,   and  then  completely  filled  with 
matter,  of  densit}^  />0.     If  c  is  the  distance  between  the  centre 
of  the  cavity  and  the  centre  of  the  sphere,  the  attraction  due 
to  the  composite  solid  at  a  point  in  the  line  joining  these  two 
centres,  at  a  distance  d  from  the  centre  of  the  sphere,  is 

4     [pa3       63(p0-p)1 
3     [_d2         (d±c)2j 

13.  The  centre  of  a  sphere  of  aluminum  of  radius  10  and  of 
density  2.5,  is  at  the  distance  100  from  a  sphere  of  the  same 


THE   ATTRACTION   OF    GRAVITATION.  27 

size  made  of  gold,  of  density  19.  Show  that  the  attraction 
due  to  these  spheres  is  nothing  at  a  point  between  them,  at  a 
distance  of  about  26.6  from  the  centre  of  the  aluminum  sphere. 

[Stone.] 

14.  Show  that  the  attraction  at  the  centre  of  a  sphere  of  radius 
r,   from   which   a  piece  has  been  cut  by  a  cone  of  revolution 
whose    vertex    is  at  the  centre,    is   irpr  siira,   where  a  is   the 
half  angle  of  the  cone. 

15.  An  iron  sphere  of  radius  10  and  density  7  has  an  eccentric 
spherical  cavity  of  radius  6,   whose   centre  is  at  a  distance  3 
from  the  centre  of    the  sphere.      Find  the  attraction   due   to 
this  solid  at  a  point  25  units  from  the  centre  of  the  sphere, 
and  so  situated  that  the  line  joining  it  with  this  centre  makes 
an  angle  of  45°  with  the  line  joining  the  centre  of  the  sphere 
and  the  centre  of  the  cavity.  [Stone.] 

16.  If  the  piece  of  a  spherical  shell  of  radii  r0  and  rlf  inter 
cepted  by  a  cone  of  revolution  whose  solid  angle  is  w  and  whose 
vertex  is  the  centre  of  the  shell,  be  cut  out  and  removed,  find 
the  attraction  of  the  remainder  of  the  shell  at  a  point  P  situated 
in  the  axis  of  the  cone  at  a  given  distance  from  the  centre  of 
the  sphere.     If  in  the  vertical  shaft  of  a  mine  a  pendulum  be 
swung,  is  there  any  appreciable  error  in  assuming  that  the  only 
matter  whose  attraction  influences  the  pendulum  lies  nearer  the 
centre  of  the  earth,   supposed  spherical,  than  the  pendulum 
does  ? 

17.  Show  that  the  attraction  of  a  spherical  segment  is,  at  its 
vertex, 


where  a  is  the  radius  of  the  sphere  and  h  the  height  of  the 
segment. 

18.    Show  that  the  resultant  attraction  of  a  spherical  segment 
on  a  particle  at  the  centre  of  its  base  is 


_ 

o  \Cl  —  nj 


28  TIIK  ATTRACTION  OF  GRAVITATION. 

19.  Show  that  the  attraction  at  the  focus  of  a  segment  of  a 
paraboloid  of  revolution  bounded  by  a  plane  perpendicular  to 
the  axis  at  a  distance  b  from  the  vertex  is  of  the  form 

T      a  4-  & 
4  IT  pa  log  —    -  — 


20.  Show  that  the  attraction  of  the  oblate  spheroid  formed 
by  the  revolution  of  the  ellipse  of  semiaxes  a,  6,  and  eccen 
tricity  e,  is,  at  the  pole  of  the  spheroid, 


ew     (.  e  j 

and  that  the  attraction  due  to  the  corresponding  prolate  spheroid 
is,  at  its  pole, 


e2  I2e      '  1-e 

21.  Show  that  the  attraction  at  the  point  (c,  0,  0),  due  to 
the  homogeneous  solid  bounded  by  the  planes  x  —  a,  x  =  b,  and 
by  the  surface  generated  by  the  revolution  about  the  axis  of  x 
of  the  curve  y=f(x},  is 


22.  Prove  that  the  attraction  of  a  uniform  lamina  in  the  form 
of  a  rectangle,  at  a  point  P  in  the  straight  line  drawn  through 
the  centre  of  the  lamina  at  right  angles  to  its  plane,  is 

ab 


sn 


where  2  a  and  2  b  are  the  dimensions  of  the  lamina  and  c  the 
distance  of  P  from    its  plane. 

[Answers  to  some  of  these  problems  and  a  collection  of  additional  prob 
lems  illustrative  of  the  text  of  this  chapter  may  be  found  near  the  end  of 
the  book.] 


THE   NEWTONIAN    POTENTIAL    FUNCTION.  29 


CHAPTER    II. 

THE   NEWTONIAN   POTENTIAL   FUNCTION   IN   THE   CASE 
OF  GEAVITATION, 

19.  Definition.  If  we  imagine  an  attracting  body  M  to  be 
cut  up  into  small  elements,  and  add  together  all  the  fractions 
formed  by  dividing  the  mass  of  each  element  by  the  distance  of 
one  of  its  points  from  a  given  point  P  in  space,  the  limit  of  this 
sum,  as  the  elements  are  made  smaller  and  smaller,  is  called  the 
value  at  P  of  "  the  potential  function  due  to  M" 

If  we  call  this  quantity  F,  we  have 

V=   lim^  ^~,  [46] 

where  A??i  is  the  mass  of  one  of  the  elements  and  r  its  distance 
from  P,  and  where  the  summation  is  to  include  all  the  elements 
which  go  to  make  up  J/. 

If  we  denote  by  p  the  average  density  of  the  element  whose 
mass  is  A??i,  and  call  the  coordinates  of  the  corner  of  this  ele 
ment  nearest  the  origin  a;',  y',  z',  and  those  of  P,  #,  y,  z,  we  may 
write 


and 

'  r  « 


=CC  C 
JJJ 


where  p  is  the  density  at  the  point  (V,  #'  ,  z')  ,  and  where  the 
triple  integration  is  to  include  the  whole  of  the  attracting  mass  M. 

As  the  position  of  the  point  P  changes,  the  value  of  the  quan 
tity  under  the  integral  signs  in  [47]  changes,  and  in  general  V 
is  a  function  of  the  three  space  coordinates,  i.e.,  V=f(x,y,z}. 

To  avoid  circumlocution,  a  point  at  which  the  value  of  the 


30  THE   NEWTONIAN   POTENTIAL   FUNCTION 

potential  function  is  Vn  is  sometimes  said  to  be  "  at  potential 
F0."  From  the  definition  of  Fit  is  evident  that  if  the  value  at 
a  point  P  of  the  potential  function  due  to  a  system  of  masses 
MI  existing  alone  is  Fi,  and  if  the  value  at  the  same  point  of 
the  potential  function  due  to  another  system  of  masses  M2  exist 
ing  alone  is  F2,  the  value  at  P  of  the  potential  function  due  to 
M!  and  M2  existing  together  is  V—  FI  +  F2. 

20.   The  Derivatives  of  the  Potential  Function.     If  P  is  a 

point  outside  the  attracting  mass,  the  quantity 


which  enters  into  the  expression  for  V  in  [47],  can  never  be 
zero,  and  the  quantity  under  the  integral  signs  is  finite  every 
where  within  the  limits  of  integration ;  now,  since  these  limits 
depend  only  upon  the  shape  and  position  of  the  attracting  mass 
and  have  nothing  to  do  with  the  coordinates  of  P,  we  may  dif 
ferentiate  Fwith  respect  to  either  x,  2/,  or  z  by  differentiating 
under  the  integral  signs.  Thus  : 


=//JW 


p  (x1  —  x]  dx'dy'dz' 


where  the  limits  of  integration  are  unchanged  by  the  differen 
tiation.  The  dexter  integral  in  this  equation  is  (Section  14) 
the  value  of  the  component  parallel  to  the  axis  of  x  of  the 
attraction  at  P  due  to  the  given  masses,  so  that  we  may  write, 
using  our  old  notation, 

AF=X,  [49] 

and,  similarly,  DyV=Y,  [50] 

DZV=Z.  [51] 

The  resultant  attraction  at  P  is 


2vy,  [52] 


IN   THE   CASE   OF   GRAVITATION.  31 

arid  the  direction-cosines  of  its  line  of  action  are  : 

and  cos  7  =  ^-^.         [53] 


Ji  H  H 

It  is  evident  from  the  definition  of  the  potential  function  that 
the  value  of  the  latter  at  any  point  is  independent  of  the  par 
ticular  system  of  rectangular  axes  chosen.  If,  then,  we  wish  to 
find  the  component,  in  the  direction  of  any  line,  of  the  attraction 
at  any  point  P,  we  may  choose  one  of  our  coordinate  axes 
parallel  to  this  line,  and,  after  computing  the  general  value  of 
F,  we  may  differentiate  the  latter  partially  with  respect  to  the 
coordinate  measured  on  the  axis  in  question,  and  substitute  in 
the  result  the  coordinates  of  P. 

21.  Theorem.  The  results  of  the  last  section  may  be  summed 
up  in  the  words  of  the  following 

THEOREM. 

To  find  the  component  at  a  point  P,  injciny  direction  PK,  oj 
the  attraction  due  to  any  attracting  mass  3/,  ice  may  divide  the 
difference  between  the  values  of  the  potential  function  due  to  M  at 
P'  (a  point  beticeen  P  and  K  on  the  straight  line  PK)  and  at  P 
by  the  distance  PP'.  The  limit  approached  by  this  fraction  as 
P'  approaches  P  is  the  component  required.* 

We  might  have  arrived  at  this  theorem  in  the  following  way  : 
If  X,  Y,  Z  are  the  components  parallel  to  the  coordinate  axes 

of  the  attraction  at  any  point  P,  the  component  in  any  direction 

PK  whose  direction  -cosines  are  A,  /u,,  and  v,  is 

XX  -f  n  Y+  rZ  =  \DX  V+  ^Dy  V+  vDz  V.  [54] 

Let  x,  y,  z  be  the  coordinates  of  P,  and  x  +  A.r,  y  -+-  Ay, 
z-\-\z  those  of  P',  a  neighboring  point  on  the  line  PK. 

*  If  the  force  is  required  in  absolute  kinetic  units,  the  result  thus 
obtained  must  be  multiplied  by  k,  the  proper  gravitation  constant.  The 
reciprocal  of  A:  is  equal  to  1.543  x  10"  in  the  c.g.s.  system  and  to  9.63  x 
108  in  the  f.p.s.  system. 


32 


THE    NEWTONIAN    POTENTIAL    FUNCTION 


If  V  and  V  are  the  values  of  the  potential  function  at  P 
and  P'  respectively,  we  have,  by  Taylor's  Theorem, 

V  =  V  +  Ax  •  I)x  V  +  Ay  •  Dv  V  +  A*  •  Dz  V  +  e, 
where  e  is  an  infinitesimal  of  an  order  higher  than  the  first. 


-  V 


pp, 

but 

therefore, 


r    I- 


r 

>  K 


1 

J 


+ 


and  this  (see  [54])  is  the  component  in  the  direction  PK  of 
the  attraction  at  P  :  which  was  to  be  proved. 

22.  The  Potential  Function  everywhere  Finite.  If  P  is  a 
point  within  the  attracting  mass,  the  integrand  of  the  expres 
sion  which  gives  the  value  of  the  potential  function  at  P 
becomes  infinite  at  P.  That  V  is  not  infinite  in  this  case  is 
easily  proved  by  making  P  the  origin  of  a  system  of  polar 
coordinates  as  in  Section  15,  when  it  will  appear  that  the 
value  of  the  potential  function  at  P  can  be  expressed  in  the 

form  rp=CCCPrBw6drd$d4>;  [57] 

and  tliis   is   evidently   finite. 

Although    VP    is    everywhere 

finite,  yet  when  we  express  its 
value  by  means  of  equation  [47], 
the  quantity  under  the  integral 
signs  becomes  infinite  within  the 
limits  of  integration,  when  P  is 
a  point  inside  the  attracting 
mass.  Under  these  circum 
stances  we  cannot  assume  with 
out  further  proof  that  the  result 

obtained  by  differentiating  with  respect  to  x  under  the  in 
tegral  signs  is  really  Dx  V.     It  is  therefore  desirable  to  com- 


ft 

r/y, 

\ 

// 

\L       > 

>^ 

« 

V 


FIG.  15. 


IN   THE   CASE   OF   GRAVITATION. 


83 


putc  the  limit  of  the  ratio  of  the  difference  (A^F)  between  the 
values  of  1'  at  the  points  /"=  (.r  -+-  Ax,  y,  z)  and  P=.(x,  y,  z), 
both  within  the  attracting  mass,  to  the  distance  (Ax)  between 
these  points.  For  convenience,  draw  through  P  (Fig.  lo)  three 
lines  parallel  to  the  coordinate  axes,  and  let  Q  =  (x'.  ?/',  z'). 


Then 


r'2  =  ?-  -h  (Ax)2—  2  r  •  Ax  •  cos  i^, 


where        cos     = 


x  —  x 


)       A* 


pdx'dy'dz' 
Ax 


Therefore 


=  CCC(  **-* 

J  J  J  ^r'r  +  rr'-, 
C  C*  /Y2?'Ax  cosi/^  —  (A.r)2\  pdx'dy'dz' 

=JJJr 


Ax 


^  j^__    limit    f^x* 


=  rrrirco^    fa,    ,   , 

JJJ  2i 

=  C  C  rpdx'dy'dz'cost, 


This  last  integral  is  evidentl}*  the  component  parallel  to  the 
axis  of  x  of  the  attraction  at  P,  so  that  the  theorem  of  Article 
21  ma}'  be  extended  to  points  within  the  attracting  mass. 

It  is  to  be  noticed  that  p  is  a  function  of  x',  ?/',  and  z',  but  not 
a  function  of  x,  y,  and  z,  and  that  we  have  really  proved  that  the 
derivatives  with  regard  to  x,  y,  and  z  of 


34 


THE   NEWTON  FAN    POTENTIAL    FUNCTION 


where  F  is  any  finite,  continuous,  and  single-valued  function  of 
a;',  yf,  and  2',  can  always  be  found  by  differentiating  under  the 
integral  signs,  whether  (x,y,z)  is  contained  within  the  limits  of 
integration  or  not. 

23.  The  Potential  Function  due  to  a  Straight  Wire.  Let 
p.  be  the  mass  of  the  unit  length  of  a  uniform  straight  wire  AB 
(Fig.  1C)  of  length  21.  Take  the  middle  point  of  the  wire  for 
the  origin  of  coordinates,  and  a  line  drawn  perpendicular  to  the 
wire  at  this  point  for  the  axis  of  x. 


y 

fy 


7 
6 

FIG.  16. 


The  value  of  the  potential  function  at  any  point  P  (x,  y)  in 
the  coordinate  plane  is,  then,  according  to  [47], 


If  r  =  AP  = 

»J2 

whence  y  = 


and    r'  =  BP  =  Vor 


4:1 

express  VP  in  terms  of  r  and  r'. 
Thus  : 


+  y)*, 

I 

,  we  may  eliminate  x  and  y  from  [59]  and 


It  is  evident  from  [60]  that  if  P  move  so  as  to  keep  the  sum 
of  its  distances  from  the  ends  of  the  wire  constant,  VP  will 


IN   THE   CASE   OF   GRAVITATION.  35 

remain  constant.     P's  locus  in  this   case  is  an  ellipse  whose 
foci  are  at  A  and  B. 
From  [59]  we  get 


x{_          r  r'         J 

r  -i 

=  £   1  —  cos  8  —  1  —  cos8f 
xl  J 

eosS'l, 


sS  +  cosS' 

and  this  (Section  6)   is  the  component  in  the  direction  of  the 
axis  of  x  of  the  attraction  at  P. 

24.  The  Potential  Function  due  to  a  Spherical  Shell.  In 
order  to  find  the  value  at  the  point  P  of  the  potential  function 
due  to  a  homogeneous  spherical  shell  of  density  p  and  of  radii  r0 
and  r1?  we  may  make  use  of  the  notation  of  Section  9. 


f  rrpi**in0drdOd<l>  _  C  C  C 


[61] 

If  P  lies  within  the  cavity  enclosed  by  the  shell,  the  limits  of 
y  are  (r  —  c)  and  (r-f-c),  whence 

F=27rp(r12-r02).  [62] 

If  P  lies  without  the  shell,  the  limits  of  y  are  (c  —  r)  and 
(c  +  v)  ?  whence 


o  C 

If  P  is  a  point  within  the  mass  of  the  shell  itself,  at  a  dis 
tance  c  from  the  centre,  we  may  divide  the  shell  into  two  parts 


THE    NEWTONIAN    POTENTIAL   FUNCTION 


by  means  of  a  spherical  surface  drawn  concentric  with  the  given 
shell  so  as  to  pass  through  P.  The  value  of  the  potential  func 
tion  at  P  is  the  sum  of  the  components  due  to  these  portions  of 
the  shell ;  therefore 

V—*rp(rt-f)+\&(t-rf) 


[G4] 

If  we  put  these  results  together,  we  shall  have  the  following 
table :  — 


F= 


3c2 


rf-rf) 


If  we  make  F,  DCF,  and  Z)c2Fthc  ordinates  of  curves  whose 
abscissas  are  c,  we  get  Fig.  17.* 

Here  LNQS  represents  F,  and  it  is  to  be  noticed  that  this 
curve  is  everywhere  finite,  continuous,  and  continuous  in  direc 
tion.  The  curve  OABC  represents  DCV.  This  curve  is  every 
where  finite  and  continuous,  but  its  direction  changes  abruptly 
when  the  point  P  enters  or  leaves  the  attracting  mass.  The 
three  disconnected  lines  OA,  DE,  and  FG  represent  DC2V. 

If  the  density  of  the  shell  instead  of  being  uniform  were  a 
function  of  the  distance  from  the  centre  [/a  =/(?•)],  we  should 
have  at  the  point  P,  at  the  distance  c  from  the  centre  of  the 
sphere, 

[65] 


*  See  Thomson  and  Tait's  Treatise  on  Natural  Philosophy.  Notice  that 
A  D  =  —  4  Ttp  and  that  EF  =  4  Ttp.  For  values  of  c  greater  than  n,  F, 
DCF,  and  DC2F  are  respectively  equal  to  M/c,  —  M/c*2,  and 
where  M  is  the  mass  of  the  shell. 


IN   THE   CASE   OF   GRAVITATION.  37 

From  this  it  follows,  as  the  reader  can  easily  prove,  that  the 
value  of  the  potential  function  due  to  a  spherical  shell  whose 
density  is  a  function  of  the  distance  from  the  centre  only  is 


constant  throughout  the  cavity  enclosed  by  the  shell,  and  at 
all  outside  points  is  the  same  as  if  the  mass  of  the  shell  were 
concentrated  at  its  centre.* 

25.  Equipotential  Surfaces.  As  we  have  already  seen,  Fis, 
in  general,  a  function  of  the  three  space  coordinates  [  V  = 
f(x,  y,  «)],  and  in  any  given  case  all  these  points  at  which  the 
potential  function  has  the  particular  value  c  lie  on  the  surface 
the  equation  of  which  is  V  =f(x,  y,  z]  =  c. 

Such  a  surface  is  called  an  "  equipotential "  or  "  level  "  sur 
face.  By  giving  to  c  in  succession  different  constant  values, 
the  equation  V  =  c  yields  a  whole  family  of  surfaces,  and  it  is 
always  possible  to  draw  through  any  given  point  in  a  field  of 
force  a  surface  at  all  points  of  which  the  potential  function  has 
the  same  value.  The  potential  function  cannot  have  two  differ 
ent  values  at  the  same  point  in  space,  therefore  no  two  differ 
ent  surfaces  of  the  family  V  =  c,  where  V  is  the  potential  func 
tion  due  to  an  actual  distribution  of  matter,  can  ever  intersect. 

*  If  the  outer  radius  of  the  shell  be  unchanged  while  the  radius  of  the 
cavity  approaches  zero,  the  values  of  V  and  DeFat  O  approach  as  limits 
the  corresponding  values  at  the  centre  of  a  solid,  homogeneous  sphere  of 
density  p  and  radius  TV  The  value  of  D0'2r,  however,  does  not  approach 
as  a  limit  the  value  of  Dj2  V  at  the  centre  of  such  a  sphere. 


38  THE   NEWTONIAN    POTENTIAL    FUNCTION 

THEOREM. 

If  there  be  any  resultant  force  at  a  point  in  space,  due  to  any 
attracting  masses,  this  force  acts  along  the  normal  to  that  equi- 
potential  surface  on  which  the  point  lies. 

For,  let  F=/(#,  y,  z)  =  c  be  the  equation  of  the  equipotential 
surface  drawn  through  the  point  in  question,  and  let  the  coordi 
nates  of  this  point  be  #0,  y0,  z0.  The  equation  of  the  plane 
tangent  to  the  surface  at  the  point  is 


and  the  direction-cosines  of  any  line  perpendicular  to  this  plane, 
and  hence  of  the  normal  to  the  given  surface  at  the  point 
Oo,  2/o,  Zo),  are 

cos  a  =  Dx*V  —  ,  [66A] 


cos/3  =  ^  [66.] 

V(A*0F)2+(A,0F)2+CD,0F)2 

and  cosy  =  -  ^QF_  -        __.  [660] 


But  if  we  denote  the  resultant  force  of  attraction  at  the  point 
(#0,  2/0,  z0)  by  R,  and  its  components  parallel  to  the  coordinate 
axes  by  X,  F,  and  Z,  these  cosines  are  evidently  equal  to 

X    Y  Z 

—  ,  —  ,  and  —  respectively,  so  that  a,  /?,  and  y  are  the  direction- 

R    R  R 

angles  not  only  of  the  normal  to  the  equipotential  surface  at  the 
point  (a?0,  2/07  ZG)  i  but  also  [35]  of  the  line  of  action  of  the  re 
sultant  force  at  the  point.  Hence  our  theorem. 

Fig.  18  represents  a  meridian  section  of  four  of  the  system 
of  equipotential  surfaces  due  to  two  equal  spheres  whose  sec 
tions  are  here  shaded.  The  value  of  the  potential  function  due 
to  two  spheres,  each  of  mass  Jf,  at  a  point  distant  respectively 
TI  and  r2  from  the  centres  of  the  spheres,  is 


IN    THE   CASE    OF   GRAVITATION. 


39 


and  if  we  give  to  V  in  this  equation  different  constant  values, 
we  shall  have  the  equations  of  different  members  of  the  system 
of  equipotential  surfaces.  Any  one  of  these  surfaces  ma}'  be 
easily  plotted  from  its  equation  by  finding  corresponding  values 


FIG.  18. 

of  rj  and  ?'2  which  will  satisfy  the  equation  ;  and  then,  with  the 
centres  of  the  two  spheres  as  centres  and  these  values  as  radii, 
describing  two  spherical  surfaces.  The  intersection  of  these 
surfaces,  if  they  intersect  at  all,  will  be  a  line  on  the  surface 
required. 

If  2  a  is  the  distance  between  the  centres  of  the  spheres, 

V  =  —  -   gives  an  equipotential  surface  shaped  like  an  hour- 

CL 

glass.  Larger  values  of  V  than  this  give  equipotential  sur 
faces,  each  one  of  which  consists  of  two  separate  closed  ovals, 
one  surrounding  one  of  the  spheres,  and  the  other  the  other. 

9  ~\r 

Values  of  Y  less  than  —  _  give  single  surfaces  which  look  more 

and  more  like  ellipsoids  the  smaller  Y  is. 

Several   diagrams    showing   the    forms  of   the   equipotential 
surfaces  due  to  different  distributions  of  matter  are  given  at 


40  THE   NEWTONIAN    POTENTIAL   FUNCTION 

the  end  of  the  first  volume  of  Maxwell's  Treatise  on  Electricity 
and  Magnetism. 

26.   The  Value  of  V  at  Infinity.     The  value,  at  the  point  P, 
of  the  potential  function  due   to   any  attracting  mass  M  has 

been  defined  to  be 

v_   limit 

~Am  =  . 

Let  r()  be  the  distance  of  the  nearest  point  of  the  attracting 
mass  from  P,  then 

[67] 


The  fraction  —  has  a  constant  numerator,  and  a  denominator 

?'o 
which  grows  larger  without  limit  the  farther  P  is  removed  from 

the  attracting  masses  ;  hence,  we  see  that,  other  things  being 
equal,  the  value  at  P  of  the  potential  function  is  smaller  the 
farther  P  is  from  the  attracting  matter  ;  and  that  if  P  be  moved 
away  indefinitely,  the  value  of  the  potential  function  at  P 
approaches  zero  as  a  limit.  In  other  words,  the  value  of  the 
potential  function  at  "infinity"  is  zero. 

About  0,  any  fixed  point  near  the  attracting  mass,  as  centre, 
imagine  a  spherical  surface,  S,  drawn,  of  fixed  radius,  r0,  so 
large  that  $  shall  just  include  all  the  distribution.  Then,  if 
P  is  any  distant  point  without  S,  and  if  OP  =  r, 

M  M  rM  rM 

or  <rVP< 


r-r          r  +  r0  r  - 


Since  --  -  r™     -7  =  1,  V  so  vanishes  at  infinity 

r  -j-  r0  T  —  PO 

that  the  limit  of  (r-  VP),  as  r  increases  without  limit,  is  M. 

Since       cos 


it  is  easy  to  see  that 

^limit  ^Dr  Y)^-M  and  that  /^  <V2DX  V)  =  -  M  cos  (a;,  r), 

where  (x,  r)  denotes  the  angle  between  the  axis  of  x  and  OP. 


IN    THE    CASE    OF    GRAVITATION.  41 

27.  The  Potential  Function  as  a  Measure  of  Work.  The 
amount  of  work  required  to  move  a  unit  mass,  concentrated  at 
a  point,  from  one  position,  P1?  to  another,  P2,  by  any  path,  in 
face  of  the  attraction  of  a  system  of  masses,  J/,  is  equal  to 


FIG.  19. 

Vi  —  V2,  where  V\  and  F2  are  the  values  at  Pl  and  P2  of  the 
potential  function  due  to  M. 

To  prove  this,  let  us  divide  the  given  path  into  equal  parts 
of  length  As,  and  call  the  average  force  which  opposes  the 
motion  of  the  unit  mass  on  its  journey  along  one  of  these 
elements  AB  (Fig.  19),  F.  The  amount  of  work  required  to 
move  the  unit  mass  from  A  to  B  is  PAs,  and  the  whole  work 
done  by  moving  this  mass  from  P1  to  P2  will  be 

limit 


As  As  is  made  smaller  and  smaller,  the  average  force  opposing 
the  motion  along  AB  approaches  more  and  more  nearly  the 
actual  opposing  force  at  A,  which  is  —DSV:  therefore 

limit 
As=C 

It  is  to  be  carefully  noticed  that  the  decrease  in  the  potential 
function  in  moving  from  Pl  to  P2  measures  the  work  required 
to  move  the  unit  mass  from  Pj  to  P2.  If  P2  is  removed  farther 
and  farther  from  J/,  F2  approaches  zero,  and  FI  —  F2  approaches 
F!  as  its  limit,  so  that  the  value  at  any  point  P1?  of  the  poten 
tial  function  due  to  any  system  of  attracting  masses,  is  equal 
to  the  work  which  would  be  required  to  move  a  unit  mass,  sup 
posed  concentrated  at  P1?  from  Pl  to  "  infinity"  by  any  path. 


42  THE    NEWTONIAN    POTENTIAL   FUNCTION 

The  work  (  W")  that  must  be  done  in  order  to  move  an  attract 
ing  mass  M'  against  the  attraction  of  any  other  mass  M,  from 
a  given  position  by  any  path  to  "  infinity/'  is  the  sum  of  the 
quantities  of  work  required  to  move  the  several  elements  (Am') 
into  which  we  may  divide  M ',  and  this  may  be  written  in  the 
form 

w  =     limit  VAm'  f  f  f pdxdydz 

Am'-O^        J  J  J  ^-.^^-(y'-y^  +  ^-zYJ 

C  C  C  C  C  C  pp'dxdydzdx'dy'dz' 

JJJJJJ[_(x'-x)2  +  (y'-y)2  +  (z'-zry' 

W  is  called  by  some  writers  "  the  potential  of  the  mass  M' 
with  reference  to  the  mass  M"  ;  by  others,  the  negative  of  W 
is  called  "  the  mutual  potential  energy  of  M  and  M' ." 

In  many  of  the  later  books  on  this  subject,  the  word  "po 
tential  "  is  never  used  for  the  value  of  the  potential  function 
at  a  point,  but  is  reserved  to  denote  the  work  required  to  move 
a  mass  from  some  present  position  to  infinity.  If  V  is  the 
value  of  the  potential  function  at  a  point  P,  at  which  a  mass 
m  is  supposed  to  be  concentrated,  m  V  is  the  potential  of  the 
mass  m.  If  we  could  have  a  unit  mass  concentrated  at  a  point, 
the  potential  of  this  mass  and  the  value  of  the  potential  function 
at  the  point  would  be  numerically  identical. 

Imagine  any  given  distribution  of  attracting  matter  which 
has  the  potential  function  V,  divided  into  elements,  of  volume 
ATI?  Ar2,  Ar3,  •  •  • ,  of  density  px,  p2,  PS?  •  •  •  >  and  of  mass  &mly 
Ara2,  Ara3,  •  •  • .  If  the  density  at  every  point  in  the  distri 
bution  were  X  times  what  it  now  is  (X  being  any  positive 
constant),  the  potential  function  would  be  XV,  and,  since  the 
volume  occupied  by  each  element  would  be  unchanged,  the 
mass  of  the  ^?th  element  would  be  XAm^.  To  change  X  to 
X  4-  AX,  the  mass  of  every  element  must  be  increased  and 
to  the  pih  element  must  be  brought  up  the  mass-increment 
AX- Amp.  If  this  quantity  were  brought  up  from  an  infinite 
distance,  the  attraction  of  the  existing  distribution  would  do 
upon  it  an  amount  of  work  represented  by  XF-AX-Aw^,  so 


IN   THE   CASE    OF    GRAVITATION.  43 

that  the  work  done  on  the  additions  to  the  whole  mass  would 
be  X  AA-  limit  \^  V  Am.  The  work  done  by  the  attractive 
forces  while  A.  was  being  changed  from  A0  to  Ax  would  be 
limit^  V  Am  •  J  \  d\.  To  find  the  work  done  by  the  attrac 

tion  for  one  another  of  its  own  parts,  while  the  given  distri 
bution  is  constructed  by  bringing  together  its  particles  from 
infinite  dispersion,  we  may  put  X0  =  0,  AI  =  1,  and  get 


where  the  summation  is  to  extend  over  the  whole  distribution. 
This  quantity,  the  negative  of  which  (when  the  matter  is 
attracting)  is  sometimes  called  "  the  intrinsic  energy  "  of  the 
distribution,  is  given  by  the  formula  in  attraction  units  of 
work.  In  absolute  kinetic  work  units, 


The  potential  function  inside  a  homogeneous  sphere  of 
radius  a  and  density  p,  at  a  distance  r  from  the  centre,  being 
2  trp  (a2  —  ^r2),  the  intrinsic  energy  of  the  sphere  is 

/*<*  -j  a  q  jif2 

Trp  (a2  -  i  r2)  4  Trpr2dr  or  — -  A2  a5  or  - 
•/i  15  5  a 

attraction  units  of  work.     If  the  c.g.s.  system  has  been  used 
throughout,  this  is  equivalent  to  go~^00)  ergs. 

If  V  and  V  are  the  potential  functions  due  to  two  neighbor 
ing  distributions,  M  and  M',  if  AJ/  and  A  M'  are  mass  elements 
of  the  two  distributions,  and  P  and  P'  points  in  A  M  and  A  J/' 
respectively,  the  mutual  potential  energy  of  M  and  J/'  may 

be  found  by  integrating  *      "  -  over  both  distributions, 

and,  since  the  order  of  integration  is  immaterial,  the  result 

may  be  written  —  CvdM'  or  —  Cv  dM. 

J  J 

The  intrinsic  energy  of  M  and  M'  considered  as  a  single  dis 
tribution  is  to  be  found  by  integrating  —  %  (  V  +  V)  over  both 


44  THE    NEWTONIAN    POTENTIAL    FUNCTION 

masses.     This  gives  —  \  CvdM  —  %  Cv'dM'  —  CvdM'  or 

the  sum  of  the  intrinsic  energies  of  M  and  M'  and  the  mutual 
energy  of  the  two. 

If  M  and  M'  were  made  up  of  matter  every  particle  of 
which  repelled  every  other  particle  according  to  the  Law  of 

Nature,  the  intrinsic  potential  energy  of  M  would  be  +  \  (  VdM 
and  the  mutual  potential  energy  of  M  and  M'  would  be 


CvdM',  or  +  Cv' 


28.  Laplace's  Equation.  We  have  seen  that  the  value  of 
the  potential  function,  and  the  component  in  any  direction  of 
the  attraction  at  the  point  P,  are  always  finite  functions  of  the 
space  coordinates,  whether  P  is  inside,  outside,  or  at  the  sur 
face  of  the  attracting  masses.  We  have  seen  also  that  by  dif 
ferentiating  V  at  any  point  in  any  direction  we  may  find  the 
always  finite  component  in  that  direction  of  the  attraction  at 
the  point.  It  follows  that  DXV,  DyV,  DZV  are  everywhere 
finite,  and  that,  in  consequence  of  this,  the  potential  function 
is  everywhere  continuous  as  well  as  finite. 

If  P  is  a  point  outside  of  the  attracting  masses,  the  quan 
tity  under  the  integral  signs  in  [48],  by  which  dx'  dy'  dz'  is 
multiplied,  cannot  be  infinite  within  the  limits  of  integration, 
and  we  can  find  D*V  by  differentiating  the  expression  for 
Dx  V  under  the  integral  signs. 

In  this  case 


p'  te'dy'd*',        [69] 
and  similarly, 

'--'-V^W         [70] 


/*  r  r 
JJJ 


3  (z1 

-  '''  [71] 


IN    THE    CASE    OF    GRAVITATION.  45 

Whence,  for  all  points  exterior  to  the  attracting  masses, 

/V  V  +  DiV  +  Dz2  V  =  0.  [72] 

This  is  Laplace's  Equation.     For  the  operator 

(Dx2  +  D;  +  A2), 

the  symbols  8,  A,  A2,  —  V2,  V2,  and  V2  have  been  used  by  dif 
ferent  authors,  and  [72]  may  be  written 

V2r  =  0.  [73] 

The  potential  function,  due  to  every  conceivable  distribu 
tion  of  matter,  must  be  such  that  at  all  points  in  empty  space 
Laplace's  Equation  shall  be  satisfied.* 

29.  The  Second  Derivatives  of  the  Potential  Function  are 
Finite  at  Points  within  the  Attracting  Mass.  If  the  point  P 
lies  within  the  attracting  mass,  V  and  DXV  are  finite,  but  the 
quantity  under  the  integral  signs  in  the  expression  for  DXV 
becomes  infinite  within  the  limits  of  integration,  and  we  can 
not  assume  that  D£V  may  be  found  by  differentiating  DXV 
under  the  integral  signs.  In  order  to  find  Dx2  V  under  these 
circumstances,  it  is  convenient  to  transform  the  equation  for 
DXV.  Let  us  choose  our  coordinate  axes  so  as  to  have  all  the 
attracting  mass  in  the  first  octant,  and  divide  the  projection  of 
the  contour  of  this  mass  on  the  plane  yz  into  elements  (dy'ds1). 
Upon  each  one  of  these  elements  let  us  erect  a  right  prism, 
cutting  the  mass  twice  or  some  other  even  number  of  times. 
Consider  one  of  the  elements  dy'dz'  the  corner  of  which  next 
the  origin  has  the  coordinates  0,  y',  and  z1.  The  prism  erected 
on  this  element  cuts  out  elements  dsl9  ds2,  dss,  ds±,  •  •  •  ds2n  from 
the  surface  of  the  attracting  mass,  and  that  edge  of  the  prism 
which  is  perpendicular  to  the  plane  yz  at  (0,  y',  2')  cuts  into 
the  surface  at  points  whose  distances  from  the  plane  of  ijz  are 
aD  as>  as>  •••  a2n-v  and  out  of  the  surface  at  points  whose  dis 
tances  from  the  same  plane  are  «2>  « *>  «e>  •  •  •  «2«-  At  every  one 

*  If  a  function,  continuous  with  its  first  derivatives  within  a  region,  T, 
satisfies  Laplace's  Equation  at  every  point  of  the  region,  it  is  sometimes 
said  to  be  harmonic  in  T. 


46 


THE    NEWTONIAN    POTENTIAL    FUNCTION 


of  these  points  of  intersection  draw  a  normal  towards  the  inte 
rior  of  the  attracting  mass,  and  call  the  angles  which  these 
normals  make  with  the  positive  direction  of  the  axis  of  x,  a1? 
a2?  as?  •  •  •  °vn*  -J-fc  ig  to  be  noticed  that  a1?  a3,  a5,  •  •  •  a2n_l  are  all 
acute,  and  that  a2,  a4,  a6,  •  •  •  a2n  are  all  obtuse.  The  element 
dy'dz'  may  be  regarded  as  the  common  projection  of  the  sur 
face  elements  ds^  ds2,  dss,  •  •  •  ds2n,  and,  so  far  as  absolute  value 
is  concerned,  the  following  equations  hold  approximately : 

dy'dz'  =  dsl  cos  ax  =  ds2  cos  a2  =  dss  cos  a3  =  •  •  •  =  ds2n  cos  a2)l. 

But  dy'dz',  dsl}  ds^,  ds3,  etc.,  are  all  positive  areas,  and  cos  a2, 
cos  a4,  cos  a6,  etc.,  are  negative,  so  that,  paying  attention  to 
signs  as  well  as  to  absolute  values,  we  have 

dy'dz'=+dsi  cos  a1=—ds2  cos  a2 =+ dss  cos  a3=— ds4  cos  a4=  etc. 


FIG.  20. 


Now 


p'(x'-x)dx'di/'dz'  _ 
—~ 


and  in  order  to  find  the  value  of  this  expression  by  the  use 
of  the  prisms  just  described,  we  are  to  cut  each  one  of  these 
prisms  into  elementary  rectangular  parallelepipeds  by  planes 
parallel  to  the  plane  of  yz  ;  we  are  to  multiply  the  values  of 
every  one  of  these  elements  which  lies  within  the  attracting 


IN   THE    CASE    OF    GRAVITATION.  47 

mass  by  the  value  of  p'Dx'  (  --  j  at  its  corner  next  the  origin 

[i.e.,  at  (x1,  ij,  2')]  ;  and  we  are  to  find  the  limit  of  the  sum 
of  these  as  dx'  is  made  smaller  and  smaller.  We  are  then  to 
compute  a  like  expression  for  each  of  the  other  prisms,  and 
to  find  the  limit  of  the  sum  of  the  whole  as  the  bases  of  the 
prisms  are  made  smaller  and  smaller  and  their  number  corre 
spondingly  increased. 

Wherever  the  function  —  is  a  continuous  function  of  x\  we 
have 


hence,  if  the  elementary  prisms  cut  the  surface  of  the  attract 
ing  mass  only  twice, 


;  [75] 

or'=al 

and,  in  general, 


c  c  ri 

+JJJrD'f'dx' 

=  lim  /    (  —  cos  ai  dsi  +  —  cos  a2  ds2  +  —  cos  as  ds3  +  •  •  • 

£—*  \  rl  T2  r3 

\ 

\  X*     >^*     /**  'I 

[77] 

where  —  is  the  value  of  the  quantity  -  at  the  point  where  the 

line  y  =  y',  z  =  z'  cuts  the  surface  of  the  attracting  mass  for 
the  kth  time,  counting  from  the  plane  yz. 

In  order  to  find  the  value  of  the  limit  of  the  sum  which 
occurs  in  this  expression,  it  is  evident  that  we  may  divide  the 
entire  surface  of  the  attracting  mass  into  elements,  multiply 


48  THE    NEWTONIAN    POTENTIAL    FUNCTION 


the  area  of  each  element  by  the  value  of  —      -  at  one  of  its 

points,  and  find  the  limit  of  the  sum  formed  by  adding  all 
these  products  together  ;    but  this  is  equivalent  to  the  surface 

integral  of  *  —    -  taken  all  over  the  outside  of  the  attracting 


mass,  so  that 


[78] 


where  the  first  integral  is  to  be  taken  all  over  the  surface  of 
the  attracting  mass  and  the  second  throughout  its  volume. 
This  expression  for  DXV  is  in  some  cases  more  convenient 
than  that  of  [48]. 

We  have  proved  this  transformation  to  be  correct,  however, 

only  when  —  is  finite  throughout  the  attracting  mass.     If  P 


r 


is  a  point  within  the  mass,  —  is  infinite  at  P.     In  this  case 

surround  P  by  a  spherical  surface  of  radius  e  small  enough  to 
make  the  whole  sphere  enclosed  by  this  surface  lie  entirely 


FIG.  21. 

within  the  attracting  mass.  This  is  possible  unless  P  lies 
exactly  upon  the  surface  of  the  attracting  mass.  Shutting 
out  the  little  sphere,  let  Vz  be  the  potential  function  due  to 
the  rest  (T2)  of  the  attracting  mass  ;  then,  since  P  is  an  out 
side  point,  with  regard  to  T2,  we  have,  by  [78], 


DxY2=f^cosa-ds'+  f^cosads+fff^dx'dy'dz',  [79] 
where  the  first  integral  is  to  be  extended  over  the  spherical 


IN    THE    CASE    OF    GRAVITATION.  49 

surface,  which  forms  a  part  of  the  boundary  of  the  attracting 
mass  to  which  Vz  is  due ;  the  second  integral  is  to  be  taken 
over  all  the  rest  of  the  bounding  surface  of  the  attracting 
mass  ;  and  the  triple  integral  embraces  the  volume  of  all  the 
attracting  mass  which  gives  rise  to  Vz. 

As  c  is  made  smaller  and  smaller,  V<>  approaches  more  and 
more  nearly  the  potential  function  J",  due  to  all  the  attract 
ing  mass. 

/-i 
—  cos  a  ds'j  cos  a  can  never  be  greater  than 

1  nor  less  than  —  1,  so  that  if  p'  is  the  greatest  value  of  p'  on 
the  surface  of  the  sphere,  the  absolute  value  of  the  integral  must 

be  less  than  ^  I  els'  or  4  -rrp't,  and  the  limit  of  this  as  €  approaches 

zero  is  zero.  The  second  integral  in  [79]  is  unaltered  by  any 
change  in  e.  If  we  make  P  the  origin  of  a  system  of  polar 
coordinates,  it  is  evident  that  the  triple  integral  in  [79]  may 
be  written 

Dx'p'  •  r  sin  0  drdOd<j>,  [SO] 


and  the  limit  which  this  approaches  as  e  is  made  smaller  and 
smaller  is  evidently  finite,  for,  if  ?•  =  0,  the  quantity  under 
the  integral  sign  is  zero. 
Therefore, 


,  r,  =  D,  l'=         cos  a  d,  +  dx'dy'ds',  [81] 

and  [79]  is  true  even  when  P  lies  within  the  attracting  mass. 
Under  the  same  conditions  we  have,  similarly, 


D9T=  |  -  cos  (3 da 

and 


DZV=\  ^cosyrfs-f-  J    I    I  -^dx'dy'dz'.         [S3] 

Observing  that  in  these  surface  integrals  r  can  never  be  zero, 
since  we  have  excluded  the  case  where  P  lies  on  the  surface 
of  the  attracting  mass,  and  that  the  triple  integrals  belong  to 


50  THE    NEWTONIAN    POTENTIAL   FUNCTION 

the  class  mentioned  in  the  latter  part  of  Section  22,  we  will 
differentiate  [81],  [82],  and  [83]  with  respect  to  x,  y,  and  z 
respectively,  by  differentiating  under  the  integral  signs.  If 
the  results  are  finite,  we  may  consider  the  process  allowable. 
Performing  the  work  indicated,  we  have 


*  V=     p'  cos  a  -  Dx         ds+ 
D*V=:  Cp'cosyDA-}ds+  C  C  CDZ(- 


and  by  making  P  the  centre  of  a  system  of  polar  coordinates 
and  transforming  all  the  triple  integrals,  it  is  easy  to  show 
that  the  values  of  7>X2F,  D/F,  D?V  here  found  are  finite, 
whether  P  is  within  or  without  the  attracting  mass,  if  the 
derivatives  of  the  density  are  finite.  This  result*  is  important. 

30.  The  Derivatives  of  the  Potential  Function  at  the  Surface 
of  the  Attracting  Mass.     Let  the  point  P  lie  on  the  surface  of 

p 


FIG.  22. 

the  attracting  mass,   or    at    some  other  surface  where  p  is 
discontinuous.     Make  P  the  centre  of  a  sphere  of  radius  e, 

*  Lejeune  Dirichlet,  Vorlesungen  iiber  die  im  umgekehrten  Verhdltniss 
des  Quadrats  der  Entfernung  wirkenden  Krdfte. 

Riemann,  Schwere,  Electricitat,  und  Magnetismus. 

It  is  to  be  noticed  that  while  the  integral  in  the  second  member  of  [48] 
represents  DXV  even  at  points  within  the  attracting  mass,  the  integral,  I, 
obtained  by  differentiating  this  expression  for  DXV  under  the  signs  of 
integration  represents  DX2F  only  at  outside  points.  Within  the  mass  I 
is  infinite,  while  DJV  is  finite. 


IX    THE    CASE    OF    GRAVITATION.  51 

and  call  the  piece  which  this  sphere  cuts  out  of  the  attracting 
mass  Tj  and  the  remainder  of  this  mass  T2.  Let  Vl  and  Vz  be 
the  potential  functions  due  respectively  to  T±  and  T2,  then 

v  =  v,  +  ra,  DX  v  =  DX  v,  +  DX  rt, 

and  the  increment  [A  (Z^  T)]  made  in  /),.  V  by  moving  from  P 
to  a  neighboring  point  P',  inside  Tj,  is  equal  to  the  sum  of  the 
corresponding  increments  \_\(DXV^)  and  A(Z>,.  r2)]  made  in 
D,F!  and  DXV* 

With  reference  to  the  space  T2,  P  is  an  outside  point,  so 
that  the  values  at  P  of  the  first  derivatives  of  T2  with  respect 
to  x,  y,  and  z  are  continuous  functions  of  the  space  coordinates 


Let  dai  be  the  solid  angle  of  an  elementary  cone  whose  vertex 
is  at  any  fixed  point  0  in  J\  used  as  a  centre  of  coordinates. 
The  element  of  mass  will  be  pr*dudr.  The  component  in  the 
direction  of  the  axis  of  x  of  the  attraction  at  0  due  to  I\  is  the 

limit  of  the  sum  taken  throughout  Tt  of  —         —  >  where  a 

is  the  cosine  of  the  angle  which  the  line  joining  0  with  the 
element  in  question  makes  with  the  axis  of  x.  The  difference 
between  the  limits  of  w  is  not  greater  than  4  TT,  and  the  differ 
ence  between  the  limits  of  r  is  not  greater  than  2  e.  If,  then, 
K  is  the  greatest  value  which  pa  has  in  Tly 

(l>.F,)0<8w* 

It  follows  from  this  that  if  P'  is  a  point  within  Tt  so  that 
PP'  <  e,  the  change  made  in  Dx  V^  by  going  from  P  to  P'  is 
far  less  than  16  TTACC  ;  but  this  last  quantity  can  be  made  as 
small  as  we  like  by  making  e  small  enough,  so  that 

limit 
PP^ 
whence 

limit     A  /  7-)  v\  _      limit     A  ,  n  rr  \   •       limit 

zb  o  A  (^  y)  ~  pp'=  o 


and  DXV  varies  continuously  in  passing  through  P.     In  a 
similar  manner,  it  may  be  proved  that  DyV  and  Dz  V  are 


52  THE    NEWTONIAN    POTENTIAL    FUNCTION 

everywhere,  even  at  places  where  the  density  is  discontinuous, 
continuous  functions  of  the  space  coordinates. 

The  results  of  the  work  of  the  last  two  sections  are  well 
illustrated  by  Fig.  17.  We  might  prove,  with  the  help  of  a 
transformation  due  to  Clausius,*  that  the  second  derivatives 
of  the  potential  function  are  finite  at  all  points  on  the  surface 
of  the  attracting  matter  where  the  curvature  is  finite,  but  that 
the  normal  second  derivatives  generally  change  their  values 
abruptly  whenever  the  point  P  crosses  a  surface  at  which  p  is 
discontinuous,  as  at  the  surface  of  the  attracting  masses.  The 
fact,  however,  that  this  last  is  true  in  the  special  case  of  a 
homogeneous  spherical  shell  suffices  to  show  that  we  cannot 
expect  all  the  second  derivatives  of  V  to  have  definite  values 
at  the  boundaries  of  attracting  bodies. 

31.  Gauss's  Theorem.  If  any  closed  surface  S  drawn  in  a 
field  of  force  be  divided  up  into  a  large  number  of  surface 


\ 


FIG.  23. 

elements,  and  if  each  one  of  these  elements  be  multiplied  by 
the  component,  in  the  direction  of  the  interior  normal  of  the 
force  of  attraction  at  a  point  of  the  element,  and  if  these 
products  be  added  together,  the  limit  of  the  sum  thus  obtained 
is  called  the  "  surface  integral  of  normal  attraction  over  S." 

If  any  closed  surface  S  be  described  so  as  to  shut  in  com 
pletely  a  mass  m  concentrated  at  a  point,  the  surface  integral 

*  Die  Potentialfunction  und  das  Potential,    §§  19-24. 


IN    THE    CASE    OF    GRAVITATION.  53 

of  normal  attraction  due  to  m,  taken  over  S,  is  4  irm  ;  and, 
in  general,  if  any  closed  surface  S  be  described  so  as  to  shut 
in  completely  any  system  of  attracting  masses  J/,  the  surface 
integral  over  S  of  the  normal  attraction  due  to  M  is  4  nM. 

In  order  to  prove  this,  divide  S  up  into  surface  elements, 
and  consider  one  of  these  ds  at  Q.  The  attraction  at  Q  in 
the  direction  QO,  due  to  the  mass  m  concentrated  at  0,  is 

=  — :•      The  component  of  this  in  the  direction  of  the 

QO2      •'•* 

in> 
interior  normal  is  —  cos  a,  and  the  contribution  which  ds  yields 

to  the  sum  whose  limit  is  the  surface   integral   required  is 

m  cos  ads  .  . 

Connect  every  point  or  the  perimeter  or  as  with 

0  by  a  straight  line,  thus  forming  a  cone  of  such  size  as  to 
cut  out  of  a  spherical  surface  of  unit  radius  drawn  about  0 
an  element  c?o>,  say.  If  we  draw  about  0  a  sphere  of  radius 
r  =  OQ,  the  cone  will  intercept  on  its  surface  an  element 
equal  to  r2  -  da>.  This  element  is  the  projection  on  the  spher 
ical  surface  of  ds;  hence  dscosa  =  r2rfa>,  approximately,  and 
the  contribution  of  the  element  ds  to  our  surface  integral  is 
mdu>.  But  an  elementary  cone  may  cut  the  surface  more  than 
once ;  indeed,  any  odd  number  of  times.  Consider  such  a 
cone,  one  element  of  which  cuts  the  surface  thrice  in  Slt  S2, 
and  $3.  Let  OS^  OS2,  and  OS3  be  called  rl5  r2,  and  r3  respec 
tively,  and  let  the  surface  elements  cut  out  of  S  by  the  cone 
be  dslf  dsz,  and  dss,  and  the  angles  between  the  line  S3O  and 
the  interior  normals  to  S  at  Slf  S2,  and  Ss  be  a1?  a2,  a3.  It 
is  to  be  noticed  that  when  the  cone  cuts  out  of  S,  the 
corresponding  angle  is  acute,  and  that  when  it  cuts  in,  the 
corresponding  angle  is  obtuse,  a!  and  a3  are  acute,  and  a2 
obtuse.  If  we  draw  about  0  three  spherical  surfaces  with 
radii  rlt  r»,  and  >-3  respectively,  the  cone  will  cut  out  of  these 
the  elements  r^t/co,  ?v</oj,  and  rs'2da).  In  absolute  size, 
dsl  =  i\2d<j)  sec^,  ds2  =  r<fd&  seca2,  and  dss  =  r3'2da>  secas, 
approximately,  but  ds2  and  r2da)  are  both  positive,  being 
areas,  and  seca2  is  negative.  Taking  account  of  sign,  then, 


54  THE    NEWTONIAN    POTENTIAL    FUNCTION 

ds2  =  —  r2du  seca2,  and  the  cone's  three  elements  yield  to  the 
surface  integral  of  normal  attraction  the  quantity 


,  . 

m    —  —  ;  —  -  H  --  —,  —  -  H  --  =—5  —  =  1  =  m  (d<*  — 
\      V  r<?  ?v      / 

However  many  times  the  cone  cuts  S9  it  will  yield  mdv  to 
the  surface  integral  required  :   all  such  elementary  cones  will 

yield  then  in  N   dw  =  m  4  TT,  if  S  is  closed,  and,  in  general,  m®, 

where  ®  is  the  solid  angle  which  S  subtends  at  0. 

If,  instead  of  a  mass  concentrated  at  a  point,  we  have  any 
distribution  of  masses,  we  may  divide  these  into  elements, 
and  apply  to  each  element  the  theorem  just  proved  ;  hence 
our  general  statement. 

If  from  a  point  0  without  a  closed  surface  S  an  elementary 
cone  be  drawn,  the  cone,  if  it  cuts  S  at  all,  will  cut  it  an  even 
number  of  times.  Using  the  notation  just  explained,  the  con 
tribution  which  any  such  cone  will  yield  to  the  surface  integral 
taken  over  S  of  a  mass  m  concentrated  at  0  is 

/  dsl  cos  ax       dsz  cos  0.0       dss  cos  as      ds±  cos  a4  \ 

m\  -  -  --  1  --  5  --  1  --  -)  --  1  --  5  --  r  •••  I 

V     V  V  *v  *v  J 

=  m  (—  dm  +  dw  —  da>  +  dot  —  •  •  •)  =  m  •  0  =  0, 
and  the  surface  integral  over  any  closed  surface  of  the  normal 
attraction  due  to  any  system  of  outside  masses  is  zero. 

The  results  proved  above  may  be  put  together  and  stated 
in  the  form  of  a 

THEOREM  DUE  TO  GAUSS. 

If  there  le  any  distribution  of  matter  partly  within  and  partly 
without  a  closed  surface  S9  and  if  M  be  the  sum  of  the  masses 
which  S  encloses,  and  M'  the  sum  of  the  masses  outside  S,  the 
surface  integral  over  S  of  the  normal  attraction  N  toward  the 
interior,  due  to  both  M  and  M',  is  equal  to  4  irM.  If  V  be  the 
potential  function  due  to  both  M  and  M',  we  have 


IN    THE    CASE    OF    GRAVITATION.  55 

It  is  easy  to  see  that  if  a  mass  M  be  supposed  concentrated 
on  any  closed  surface  S  the  curvature  of  which  is  everywhere 
finite,  the  surface  integral  of  normal  attraction  taken  over  S 
will  be  2  TrM;  for  all  the  elementary  cones  which  can  be  drawn 
from  a  point  P  on  the  surface  so  as  to  cut  S  once  or  some 
other  odd  number  of  times,  lie  on  one  side  of  the  tangent  plane 
at  the  point,  and  intercept  just  half  the  surface  of  the  sphere 
of  unit  radius  the  centre  of  which  is  P. 

From  Gauss's  Theorem  it  follows  immediately  that  at  some 
parts  of  a  closed  surface  situated  in  a  field  of  force,  but  en 
closing  none  of  the  attracting  mass,  the  normal  component  of 
the  resultant  attraction  must  act  towards  the  interior  of  the 
surface  and  at  some  parts  toward  the  exterior,  for  otherwise 
the  limit  of  the  sum  of  the  intrinsically  positive  elements  of 
the  surface,  each  one  multiplied  by  the  component  in  the 
direction  of  the  interior  normal  of  the  attraction  at  one  of  its 
own  points,  could  not  be  zero.  In  other  words,  the  potential 
function,  the  rate  of  change  of  which  measures  the  attraction, 
must  at  some  parts  of  the  surface  increase  and  at  others 
decrease  in  the  direction  of  the  interior  normal. 

32.  Tubes  of  Force.  A  line  which  cuts  orthogonally  the  dif 
ferent  members  of  the  system  of  equipotential  surfaces  cor 
responding  to  any  distribution  of  matter  is  called  a  "line  of 
force,"  since  its  direction  at  each  point  of  its  course  shows  the 
direction  of  the  resultant  force  at  the  point.  If  through  all 
points  of  the  contour  of  a  portion  of  an  equipotential  surface 
lines  of  force  be  drawn,  these  lines  lie  on  a  surface  called  a 


FIG.  24. 


••tube  of  force."  We  may  easily  apply  Gauss's  Theorem  to  a 
space  cut  out  and  bounded  by  a  portion  of  a  tube  of  force  and 
two  equipotential  surfaces  ;  for  the  sides  of  the  tube  do  not  con- 


56  THE   NEWTONIAN    POTENTIAL   FUNCTION 

tribute  anything  to  the  surface  integral  of  normal  attraction,  and 
the  resultant  force  is  all  normal  at  points  in  the  equipotential 
surfaces.  If  w  and  w'  are  the  areas  of  the  sections  of  a  tube  of 
force  made  by  two  equipotential  surfaces,  and  if  F  and  F'  are 
the  average  interior  forces  on  w  and  w',  we  have 

Fu+F'u>'  =  0  [87] 

if  the  tube  encloses  ernpt}"  space,  and 

F<»-\-F'w'=±7rm  [88] 

when  the  tube  encloses  a  mass  in  of  attracting  matter. 

33.  Spherical  Distributions.  In  the  case  of  a  distribution 
about  a  point  in  spherical  shells,  so  that  the  density  is  a 
function  of  the  distance  from  this  point  only,  the  lines  of  force 
are  straight  lines  whose  directions  all  pass  through  the  central 
point.  Every  tube  of  force  is  conical,  and  the  areas  cut  out  of 
different  equipotential  surfaces  by  a  given  tube  of  force  are  pro 
portional  to  the  square  of  the  distance  from  the  centre. 

Consider  a  tube  of  force  which  intercepts  an  area  ^  from  a 
spherical  surface  of  unit  radius  drawn  with  0  as  a  centre,  and 
apply  Gauss's  Theorem  to  a  box  cut  out  of  this  tube  by  two 
equipotential  surfaces  of  radii  r  and  (r-j-Ar)  respectively. 


Let  AOB  (Fig.  25)  be  a  section  of  the  tube  in  question. 
The  area  of  the  portion  of  spherical  surface  <o  which  is  repre 
sented  in  section  at  ad  is  ?*2^,  and  the  area  of  that  at  be  is 
(r-|-  Ar)2  \l/.  If  the  average  force  acting  on  w  toward  the  inside 
of  the  box  is  F.  the  average  force  acting  on  w'  toward  the  inside 
of  the  box  will  be  —  (F  +  Ar7^),  and  the  surface  integral  of 
normal  attraction  taken  all  over  the  outside  of  the  box  is 

)  (r  +  Ar)  V  =  -^ .  \(F-  r2)          [89] 


IN    THE   CASE   OF   GRAVITATION.  57 

If  the  tube  of  force  which  we  have  been  considering  be  ex 
tended  far  enough,  it  will  cut  all  the  concentric  layers  of  matter, 
traverse  all  the  empty  regions  between  the  layers,  if  there  are 
such,  and  finally  emerge  into  outside  space. 

If  we  choose  r  so  that  the  box  shall  contain  no  matter,  the 
surface  integral  taken  over  the  box  must  be  zero. 

In  this  case, 


therefore,  F=  -2,  [90] 

and  V=--+IL.  [91] 

From  this  it  follows  that  in  a  region  of  empty  space,  either 
included  between  the  two  members  of  a  system  of  concentric 
spherical  shells  of  density  depending  only  upon  the  distance 
from  the  centre,  or  outside  the  whole  system,  the  force  of  attrac 
tion  at  different  points  varies  inversely  as  the  squares  of  the 
distances  of  these  points  from  the  centre. 

Suppose  that  the  box  (abed)  lies  in  a  shell  whose  densit}'  is 
constant  ;  then  the  surface  integral  of  normal  attraction  taken 
over  the  box  is  equal  to  4?r  times  the  matter  within  the  box.  In 
this  case  the  quantity  of  matter  inside  the  box  is 

-r']!-   or    p^Ar  +  e, 
4?r 

where  e  is  an  infinitesimal  of  an  order  higher  than  the  first. 
Therefore, 


whence  F=--+,  [92] 

O  ?*"" 

and  F=---^7r/0r'  +  />t.  [93] 


58  THE   NEWTONIAN   POTENTIAL  FUNCTION 

If  the  box  lies  in  a  shell  whose  density  is  inversely  propor 
tional  to  the  distance  from  the  centre,  we  shall  have 


whence  F=-'>Tr\  +     ,  [95] 


and  V=  -  -  -  '2  7r\r  -f  p.  [96] 

In  general,  if  the  box  lies  in  a  shell  whose  density  is/(r),  we 
shall  have 


'  [97] 

whence  F=  -    -  ~f(r)^  •  dr.  [98] 

In  order  to  learn  how  to  use  the  results  just  obtained  to  de 
termine  the  force  of  attraction  at  any  point  due  to  a  given 
spherical  distribution,  let  us  consider  the  simple  case  of  a  single 
shell,  of  radii  4  and  5,  and  of  density  [Ar]  proportional  to  the 
distance  from  the  centre. 

At  points  within  the  cavity  enclosed  by  the  shell  we  must 
have,  according  to  [90]  and  [91], 

F=^     and     F=  -£+/*; 
r  r 

But  the  force  is  evidently  zero  at  the  centre  of  the  shell,  where 
r  is  zero,  so  that  c  must  be  zero  everywhere  within  the  cavity, 
and  there  is  no  resultant  force  at  any  point  in  the  region.  The 
value,  at  the  centre,  of  the  potential  function  due  to  the  shell  is 
evidently 

244  TT\  roon 

-  ,  [99] 

O 

and  it  has  the  same  value  at  all  other  points  in  the  cavity. 

In  the  shell  itself  it  is  easy  to  see  that  we  must  have  at  all 
points, 


IN    THE   CASE    OF    GRAVITATION.  59 

In  order  to  determine  the  constants  in  this  equation,  we  may 
make  use  of  the  fact  that  F  and  V  are  even-where  continuous 
functions  of  the  space  coordinates,  so  that  the  values  of  .F  and 
V  obtained  by  putting  r  =  4,  the  inner  radius  of  the  shell,  in 
[100],  must  be  the  same  as  those  obtained  by  making  r  =  4  in 
the  expressions  which  give  the  values  of  F  and  V  for  the  cavity 
enclosed  by  the  shell.  This  gives  us 

•       ^~n  T        ,      500  TrA 

c'  =  2o67rA     and     //.  =  -     —  , 

o 

so  that  for  points  within  the  mass  of  the  shell  we  have 


and 


F=  -TrAr'.  [101] 


For  points  without  the  shell  we  have  the  same  general  expres 
sions  for  F  and  V  as  for  points  within  the  cavity  enclosed  by 
the  shell,  namely, 

F=\     and     F=--  +  m,  [103] 

but  the  constants  are  different  for  the  two  regions. 

Keeping  in  mind  the  fact  that  F  and  V  are  continuous,  it  is 
easy  to  see  that  we  must  get  the  same  result  at  the  boundary  of 
the  shell,  where  r  —  5.  whether  we  use  [103],  or  [101]  and  [102]. 

This  gives 

k  =  —  3G9  TrA     and     m  =  0  ; 

so  that  for  all  points  outside  the  shell  we  have 


and  V=  .  [105] 

These  last  results  agree  with  the  statements  made  in  Section 
13,  for  the  mass  of  the  shell  is  369  TrA. 

The  values,  at  every  point  in  space,  of  the  potential  function 
and  of  the  attraction  due  to  any  spherical  distribution  may  be 


60  THE   NEWTONIAN    POTENTIAL    FUNCTION 

found  by  determining,  first,  with  the  aid  of  Gauss's  Theorem, 
the  general  expressions  for  F  and  V  in  the  several  regions  ; 
then  the  constants  for  the  innermost  region,  remembering  that 
there  is  no  resultant  attraction  at  the  centre  of  the  system  ;  and 
finally,  in  succession  (moving  from  within  outwards) ,  the  con 
stants  for  the  other  regions,  from  a  consideration  of  the  fact 
that  no  abrupt  change  in  the  values  of  either  F ov  Vis  made  by 
crossing  the  common  boundary  of  two  regions. 

This  method  of  treating  problems  is  of  great  practical  im 
portance. 

34.  Cylindrical  Distributions.  In  the  case  of  a  cylindrical 
distribution  about  an  axis,  where  the  density  is  a  function  of 
the  distance  from  the  axis  only,  the  equi potential  surfaces  are 
concentric  cylinders  of  revolution  ;  the  lines  of  force  are  straight 
lines  perpendicular  to  the  axis  ;  and  every  tube  of  force  is  a 
wedge. 

If  we  apply  Gauss's  Theorem  to  a  box  shut  in  between  two 
equipotential  surfaces  of  radii  r  and  r  -f  Ar,  two  planes  perpen 
dicular  to  the  axis,  and  two  planes  passing  through  the  axis, 


FIG.  26. 

we  have,  if  \j/  is  the  area  of  the  piece  cut  out  of  the  cylindrical 
surface  of  unit  radius  by  our  tube  of  force, 

o>  =  r-i/A,      to'  =  (r-\-  Ar) •  i/', 
and  for  the  surface  integral  of  normal  attraction  taken  over  the 

box, 

F«-h>V  =  -^-Ar(r.jFf).  [10G] 

If  our  box  is  in  empty  space, 

Ar(r.F)  =  0, 

so  that  F={'     lind      V  =  clo ' 


IN   THE   CASE   OF   GRAVITATION. 


61 


If  the  box  is  within  a  shell  of  constant  density  p, 
so  that       F=-  —  ~2-pr     and     Y=c\o^r  — 


[108] 


35.  Poisson's  Equation.  Let  us  now  apply  Gauss's  Theorem 
to  the  ease  where  our  closed  surface  is  that  of  an  element  of 
volume  of  an  attracting  mass  in  which  p  is  either  constant  or  a 
continuous  function  of  the  space  coordinates.  We  will  consider 
three  cases,  using  first  rectangular  coordinates,  then  cylinder 
coordinates,  and  finally  spherical  coordinates. 


I.  In  the  first  case,  our  element  is  a  rectangular  parallelepiped 
(Fig.  27).  Perpendicular  to  the  axis  of  x  are  two  equal  sur 
faces  of  area  Ay  -  Az,  one  at  a  distance  x  from  the  plane  yz,  and 
one  at  a  distance  x  +  A#  from  the  same  plane.  The  average 
force  perpendicular  to  a  plane  area  of  size  AyAz,  parallel  to  the 
plane  yz,  and  with  edges  parallel  to  the  axes  of  y  and  z,  is  evi 
dently  some  function  of  the  coordinates  of  the  corner  of  the 
element  nearest  the  origin. 

That  is,  if  P=(x,  ?/,  z),  the  average  force  on  PP4  parallel  to 
the  axis  of  x  is  X=f(x.  y,  z),  and  the  average  force  on  P1P7  in 
the  same  direction  is  /(x  +  Aa;,  y,  z)  =  X  +  AXX,  so  that  PP4 
and  P1P7  yield  towards  the  surface  integral  of  interior-normal 

attrition  taken  over  the  element,  the  quantity  —  A-t-AyAz 


A.r 


. 
Similarly,    the   other  two   pairs  of   elementary  surfaces  yield 


62 


THE  NEWTONIAN   POTENTIAL   FUNCTION 


—  A#A?/Az—  2—  and  —  A#  A?/  Az  —  £—  ,  and,  if  p0  is  the  average 
A?/  Az 

density  of  the  matter  enclosed  by  the  box,  we  have 


s.     [100] 


This  equation  is  true  whatever  the  size  of  the  element  A#  A?/  Az. 
If  this  element  is  made  smaller  and  smaller,  the  average  nor 
mal  force  [X]  on  Pl\  approaches  in  value  the  force  \_DXV~\  at 
P  in  the  direction  of  the  axis  of  x  ;  Y  and  Z  approach  respec 
tively  the  limits  DyV  and  DZV',  and  p0  approaches  as  its  limit 
the  actual  density  \_p]  at  P. 

Taking  the  limits  of  both  sides  of  [109],  after  dividing  by 
AxA?/Az,  we  have 

A2  V+D*V+D,*V=-±*P, 
or  V2F=-47T/3,  [110] 

which  is  Poisson's  Equation.  The  potential  function  due  to  any 
conceivable  distribution  of  attracting  matter  must  be  such  that 
at  all  points  within  the  attracting  mass  this  equation  shall  be 
satisfied. 

For  points  in  empty  space  p  =  0,  and  Poisson's  Equation 
degenerates  to  Laplace's  Equation. 

II.  In  the  case  of  cylindrical  coordinates,  the  element  of  vol 
ume  (Fig.  28)  is  bounded  by  two  cylindrical  surfaces  of  revo- 


lution  having  the  axis  of  z  as  their  common  axis  and  radii  r  and 
/•  -f-  Ar,  two  planes  perpendicular  to  this  axis  and  distant  Az 


IX    THE   CASE   OF   GRAVITATION. 


63 


from  each  other,  and  two  planes  passing  through  the  axis  and 
forming  with  each  other  the  diedral  angle  A0. 

Call  Jf?,  0,  and  Z  the  average  normal  forces  upon  the  elemen 
tary  planes  PP6,  PP2,  and  PP3  respectively,  then  the  surface 
integral  of  normal  attraction  over  the  volume  element  will  be 

—  A0Az  Ar(r-  R}  —  \r\z\Q  —  A0  \_r\r  +  ^(A/-)2]ACZ 

=  47rp0  (vol.  of  box)  ;  [111] 

whence,  approximately, 

1  A  /~z?\      1   x  ^        v    7  vol. of  box     ril9-. 


Ar 


/•   A<9 


Az 


The  force  at  Pin  direction  PP5  is  DJ7,  in  direction  PP4  is  DZV, 
and  perpendicular  to  LP  in  the  plane  PLPl  is  -  •  DQ  T7,  so  that 
if  the  box  is  made  smaller  and  smaller,  our  equation  approaches 

tlieform     lA(r.AF)+iD/F+A'F=-4^.         [113] 


FIG.  29. 

III.    In  the  case  of  spherical  coordinates,  the  volume  element 
is  of  the  shape  shown  in  Fig.  29.     Let  OP=r,  ZOP=0,  and 


64  THE    NEWTONIAN    POTENTIAL    FUNCTION 

denote  by  </>  the  diedral  angle  between  the  planes  ZOP  and 
*ZOX.  Denote  by  7t,  ©,  and  <l>  the  average  normal  forces  on  tlie 
faces  jP/e,  /V's,  and  Pl\  respectively  ;  then  the  surface  integral 
of  normal  attraction  over  the  elementary  box  is  approximately 


whence         -  -  + 


=  47r/v(vol.  of  box)  ; 
** 


-         A?-           r  sin  6     A</>       r  sin  0  A0 

vol.  of  box  m  -n 

=  —  47TA,'—  -                                     --  1*15  1 

0     - 


The  force  at  Pin  the  direction  PPS  is  7)rF,  in  the  direction 

PPl  is  —I—  -DA  T7,  and  in  the  direction  PP4  is  i-DaF;  there- 
9'  sin  0  r 

fore,  as  the  element  of  volume  is  made  smaller  and  smaller,  our 
equation  approaches  the  form 

sin  0  -  Dr(r-Dr  V)  +  D^V  +  De  (sin  0  •  De  V) 


This  equation,  as  well  as  that  for  cylinder  coordinates,  might 
have  been  obtained  by  transformation  from  the  equation  in 
rectangular  coordinates. 

We  may  devote  the  rest  of  this  section  to  the  stating  of 
some  general  results  which  will  be  intelligible  only  to  those 
readers  who  are  familiar  with  the  theory  and  the  use  of 
curvilinear  coordinates. 

If  «,  v,  w  are  any  three  analytic  functions  of  x,  y,  z  which 
define  a  set  of  orthogonal  curvilinear  coordinates,  and  if 


hw*  =  (^Dxw)*  -\-  (I)yw)*  +  (Dzw)*9  it  is  possible  to   show  that 
Poisson's  Equation  may  be  written  in  either  of  the  forms 

DM2  V-  li*  +  D*  V-  h*  +  &„*  V-  lij  +  DUV.  v2^ 
+  D  V-  V2^  + 


IN    THE    CASE    OF    GRAVITATION.  65 


h...ht.-h.A  DA  —. 


By  giving  to  c  in  the  equation  u  =  c,  where  u  is  a  given 
function  of  (x,  y,  s),  different  values  in  succession  we  may  get 
the  equations  of  any  number  of  surfaces  on  each  of  which  u 
is  constant.  These  surfaces  may  or  may  not  be  the  equipo- 
tential  surfaces  of  a  possible  distribution  of  matter.  If  they 
are,  it  must  be  possible  to  find  a  potential  function  which 
changes  only  when  u  changes  and  is,  therefore,  a  function  of 
u  only.  We  may  in  this  case  consider  u  as  one  of  a  set  of 
three  orthogonal  curvilinear  coordinates  (u,  v,  w),  and  since, 
by  hypothesis,  Dv  V  =  0,  and  Dw  V  =  0,  we  may  write  Laplace's 
Equation  in  the  form  D*V-  h*  +  DUV •  V2*«  =  0,  or 


If  now  the  ratio  of  V'2^  to  h*  is  expressible  as  a  function  of 
u  only,  the  equation  is  an  ordinary  differential  equation  the 
solution  of  which  gives  the  most  general  solution  of  Laplace's 
Equation  which  is  a  function  of  u  only.  If,  however,  the 
ratio  of  \~a  to  hu2  is  not  expressible  as  .a  function  of  u  only,  V, 
which  by  hypothesis  involves  u  only,  must  satisfy  a  differ 
ential  equation  which  involves  besides  u  one  or  both  of  the 
other  coordinates  v  and  ir,  so  that  we  infer  that  no  solution  of 
Laplace's  Equation  exists  which  is  everywhere  a  function  of 
u  only.  A  set  of  confocal  ellipsoidal  surfaces  forms  a  possible 
set  of  equipotential  surfaces,  while  a  family  of  concentric,  sim 
ilar,  and  similarly  placed  ellipsoidal  surfaces  cannot  be  the 
level  surfaces  in  empty  space  of  any  distribution  of  matter. 
Two  concentric,  similar,  and  similarly  placed  ellipsoidal  sur 
faces,  Sl  and  S21  may  be  equipotential  but,  in  this  case,  the 
level  surfaces  between  Sl  and  S2  will  not  be  ellipsoidal  sur 
faces  similar  to  them. 


66  THE    NEWTONIAN    POTENTIAL    FUNCTION 

36.  Poisson's  Equation  in  the  Integral  Form.  In  [109]  A' 
may  be  regarded  as  a  function  of  x,  ?/,  z,  A//,  and  Az,  which  ap 
proaches  Z^Fas  a  limit  when  A?/  and  Az  are  made  to  approach 
zero,  and  it  may  not  be  evident  that  the  limit,  when  A.T,  Ay,  and 

A  ~v 
Az  are  together  made  to  approach  zero,  of  the  fraction   -^--  is 

Z>/F.  For  this  reason  it  is  worth  while  to  establish  Poisson's 
P^quation  by  another  method. 

It  is  shown  in  Section  29  that  the  volume   integral  of  the 

quantity  —  Dx(  -  1,  taken  throughout  a  certain  region,  is  the  sur 
face  integral  of  ^cosa  taken  all  over  the  surface  which  bounds 
r 

the  region.     In  this  proof  we  might  substitute  for  -  any  other 

function  of  the  three  space  coordinates  which  throughout  the 
region  is  finite,  continuous,  and  single-valued,  and  state  the 
results  in  the  shape  of  the  following  theorem  : 

If  T  is  an}'  closed  surface  and  U  a  function  of  oj,  ?/,  and  z 
which  for  every  point  inside  T  has  a  finite,  definite  value  which 
changes  continuously  in  moving  to  a  neighboring  point,  then 

f  f  fDJJ-dxdydz  =-  Cu  cos  ads,  [117] 

f  f  ( DyU'dxdydz=-  Cucosflds,  [118] 

and  f  f  CDZ  U-  dx  dy  dz  =  -  Cucos  y  ds,  [119] 

where  a,  /?,  and  y  are  the  angles  made  by  the  interior  normals 
at  the  various  points  of  the  surface  with  the  positive  direction 
of  the  coordinate  axes,  and  where  the  sinister  integrals  are  to  be 
extended  all  through  the  space  enclosed  by  T,  and  the  dexter 
integrals  all  over  the  bounding  surface. 

If  we  apply  this  theorem  to  an  imaginary  closed  surface  which 
shuts  in  any  attracting  mass  of  density  either  uniform  or  vari 
able,  and  if  for  £7 in [11 7],  [118], and  [119]  we  use  respectively 


IN   THE    CASE    OF    GRAVITATION.  67 

Dx  V,  I>y  V,  and  Dz  V,  and  add  the  resulting  equations  together, 
we  shall  have 


*2  V  +  P*  V  +  A'  O  dxdydz 
~j(D*v  cos  a  +  D,f  V  cos  ft  4-  A  V  cos  7)  <&.      [120] 

The  integral  in  the  second  member  of  this  equation  is  evi 
dently  (see  [56])  the  surface  integral  of  normal  attraction 
taken  over  our  imaginary  closed  surface,  and  this  by  Gauss's 
Theorem  is  equal  to  4  TT  times  the  quantity  of  matter  inside 
the  surface,  so  that 

V+  Di  V+  />/  H  dxdydz 

[121] 


Since  this  equation  is  true  whatever  the  form  of  the  closed 
surface,  we  must  have  at  every  point 


For  if  throughout  any  region  V2Fwere  greater  than  -47rp,  we 
might  take  the  boundary  of  this  region  as  our  imaginary  surface. 
lu  this  case  every  term  in  the  sum  whose  limit  gives  the  sinister 
of  [121]  would  be  greater  than  the  corresponding  term  in  the 
dexter,  so  that  the  equation  would  not  be  true.  Similar  reason 
ing  shuts  out  the  possibility  of  V2F's  being  less  than  —  47rp. 

37.  The  Average  Value  of  the  Potential  Function  on  a  Spheri 
cal  Surface.  If,  in  a  field  of  force  due  to  a  mass  m  concentrated 
at  a  point  P,  we  imagine  a  spherical  surface  to  be  drawn  so  as 
to  exclude  P,  the  surface  integral  taken  over  this  surface  of  the 
value  of  the  potential  function  due  to  m  is  equal  to  the  area  of 
the  surface  multiplied  by  the  value  of  the  potential  function  at 
the  centre  of  the  sphere. 

To  prove  this,  let  the  radius  of  the  sphere  be  a  and  the  dis 
tance  [OP]  of  P  from  its  centre  c.  Take  the  centre  of  the 


68  THE    NEWTONIAN    POTENTIAL    FUNCTION 

sphere  for  origin  and  the  line  OP  for  the  axis  of  x.  Divide  the 
surface  of  the  sphere  into  zones  by  means  of  a  series  of  planes 
cutting  the  axis  of  x  perpendicularly  at  intervals  of  Ax.  The 
area  of  each  one  of  these  zones  is  2  TTCL  dx,  so  that  the  surface 

i    £  m  • 
integral  of  --  is 


/+  a       m  2  ira  dx  |~2  irma  Va2  +  c2  —  2  cx~\ 

-«    Va2  +  c2  -  2  ex  L  G  J-a 

and  the  value  of  this,  since  the  radical  represents  a  positive 


.  .  ... 

quantity,  is  --  >  which  proves  the  proposition. 

0 

The  surface  integral  of  the  potential  function  taken  over  the 
sphere,  divided  by  the  area  of  the  sphere,  is  often  called  "  the 
average  value  of  the  potential  function  on  the  spherical  surface." 

If  we  have  any  distribution  of  attracting  matter,  we  may 
divide  it  into  elements,  apply  the  theorem  just  proved  to  each 
of  these  elements,  and,  since  the  potential  function  due  to  the 
whole  distribution  is  the  sum  of  those  due  to  its  parts,  assert 
that: 

The  average  value  on  a  spherical  surface  of  the  potential  func 
tion  due  to  any  distribution  of  matter  entirely  outside  the  sphere 
is  equal  to  the  value  of  the  potential  function  at  the  centre  of  the 
sphere. 

If  a  function,  U,  of  the  space  coordinates  attains  a  maxi 
mum  (or  a  minimum)  value  at  a  point,  Q,  it  is  possible  to  draw 
about  Q  as  centre  a  spherical  surface,  S,  of  radius  so  small 
that  the  value  of  U  at  every  point  of  S  shall  be  less  (or 
greater)  than  the  value  of  U  at  Q.  It  follows,  therefore, 
from  the  theorem  just  stated  that  : 

The  potential  function  due  to  a  finite  distribution  of  matter 
cannot  attain  either  a  maximum  or  a  minimum  value  at  any 
point  in  empty  space. 

We  may  infer  from  the  first  of  the  theorems  just  stated 
that,  if  the  potential  function  is  constant  within  any  closed 
surface,  S,  drawn  in  a  region,  T,  which  contains  no  matter,  it 


IN    THE    CASE    OF    GRAVITATION.  69 

will  have  the  same  value  in  those  parts  of  T  which  lie  outside 
S.  For,  if  the  values  of  the  potential  function  at  points  in 
empty  space  just  outside  S  were  different  from  the  value  in 
side,  it  would  always  be  possible  to  draw  a  sphere  of  which 
the  centre  should  be  inside  S,  and  which  outside  S  should  in 
clude  only  such  points  as  were  all  at  either  higher  or  lower  po 
tential  than  the  space  inside  S ;  but  in  this  case  the  value  of 
the  potential  function  at  the  centre  of  the  sphere  would  not  be 
the  average  of  its  values  over  its  surface.  A  more  satisfac 
tory  proof  can  be  given  with  the  help  of  Spherical  Harmonics. 

The  value  of  the  potential  function  cannot  be  constant  in 
unlimited  empty  space  surrounding  an  attracting  mass  J/,  for, 
if  it  were,  we  could  surround  the  mass  by  a  surface  over 
which  the  surface  integral  of  normal  attraction  would  be  zero 
instead  of  4  -n-M. 

The  average  value  on  a  spherical  surface  of  the  potential 
function  [  V~\,  due  to  any  distribution  [J/]  of  attracting  matter 
wholly  within  the  surface,  is  the  same  as  if  M  were  concen 
trated  at  the  centre  0  of  the  space  which  the  surface  encloses. 
For  the  average  values  [  F0  and  F0  +  A,.F0]  of  V  on  con 
centric  spherical  surfaces  of  radii  r  and  r  +  Ar  may  be  written 

I  Vds  (or   —  I  Fc?<o,  if  do)  is  the  solid  angle  of  an  ele- 

47r>    */  \irJ 

meutary  cone  with  vertex  at  O,  which  intercepts  the  element  ds 

1    C 

from  the  surface  of  a  sphere  of  radius  ?•),  and  -     I  (F+  Ar  V)dw  ; 

47T*/ 

whence  Ar  F0  =  --  (  A,  F-  efo>, 

and  Z>PF0  =  — 

\ir 

Now  —  \  DrV-a?du  is  the  integral  of  normal  attraction  taken 
over  the  spherical  surface,  whence,  by  Gauss's  Theorem, 

and 
7r?- 

since  VU  =  Q,  for  r=oo. 


70  THE   NEWTONIAN    POTENTIAL   FUNCTION 

38.  The  Equilibrium  of  Fluids  at  Rest  under  the  Action  of 
Given  Forces.  Elementary  principles  of  Hydrostatics  teach  us 
that  when  an  incompressible  fluid  is  at  rest  under  the  action  of 
any  system  of  applied  forces,  the  hydrostatic  pressure  p  at  the 
point  (x,  ?/,  z)  must  satisfy  the  differential  equation 

dp  =  P(Xdx  +  Ydy  +  Zdz)  ,  [122] 

where  X,  F,  and  Z  are  the  values  at  that  point  of  the  force 
applied  per  unit  of  mass  to  urge  the  liquid  in  directions  parallel 
to  the  coordinate  axes. 

For,  if  we  consider  an  element  of  the  liquid  [Ax  A?/  As] 
(Fig.  27)  whose  average  density  is  p0  and  whose  corner  next 
the  origin  has  the  coordinates  (x,  y,  z)  ,  and  if  we  denote  by  px 
the  average  pressure  per  unit  surface  on  the  face  PP%P±P^  by 
px  +  &xpx  the  average  pressure  on  the  face  P1P5P7P6,  and  by 
Xn  the  average  applied  force  per  unit  of  mass  which  tends  to 
move  the  element  in  a  direction  parallel  to  the  axis  of  x,  we 
have,  since  the  element  is  at  rest, 

-f  p0X0  Ax  A?/Az  =  (px  +  Axpx)  A?/  As, 


If  the  element  be  made  smaller  and  smaller,  the  first  side  of 
the  equation  approaches  the  limit  pX,  and  the  second  side  the 
limit  Dxp,  where  p  is  the  hydrostatic  pressure,  equal  in  all  direc 
tions,  at  the  point  P. 

This  gives  us  Dxp  =  p  X.  [123] 

In  a  similar  manner,  we  may  prove  that 

D,P  =  pY, 

and  Dzp  =  p  Z  ; 

whence      dp  =  Dxp  dx  -f  Dyp  dy  +  Dzp  dz 


If  in  any  case  of  a  liquid  at  rest  the  only  external  force 
applied  to  each  particle  is  the  attraction  due  to  some  outside 
mass,  or  to  the  other  particles  of  the  liquid,  or  to  both  together, 
X,  F,  and  Z  are  the  partial  derivatives  with  regard  to  x,  ?/,  and 


IN    THE    CASE    OF    GRAVITATION.  71 

sots,  single  function  V,  and  we  may  write  our  general  equation 
in  the  form 

dp  =  p  (DXV-  dx  +  DVV>  dy  +  Dz  V-dz)  =  P-dV, 
whence,  if  p  is  constant, 

p  =  p  V  +  const.,  [I24] 

and  the  surfaces  of  equal  hydrostatic  pressure  are  also  equi- 
potential  surfaces. 

According  to  this,  the  free  bounding  surfaces  of  a  liquid  at 
rest  under  the  action  of  gravitation  only  are  equipotential. 

EXAMPLES. 

1.  Prove  that  a  particle  cannot  be  in  stable  equilibrium 
under  the  attraction  of  any  system  of  masses.     [Earnshaw.] 

2.  The  earth's  potential  function  expressed  in  the  common, 
kinetic,  centimetre-gramme-second  units  is  981  «2/r,  for  points 
above  the  surface. 

3.  Prove  that  if  all  the  attracting  mass  lies  within  an  equi 
potential  surface  £  on  which  V  =  C,  then  in  all  space  outside 
S  the  value  of  the  potential  function  lies  between  C  and  0. 

4.  The  source  of  the  Mississippi  River  is  nearer  the  centre  of 
the  earth  than  the  mouth  is.     What  can  be  inferred  from  this 
about  the  slope  of  level  surfaces  on  the  earth  ? 

5.  If  in  [59]  x  be  made  equal  to  zero,  V  becomes  infinite. 
How  can  you  reconcile  this  with  what  is  said  in  the  first  part  of 
Section  22? 

6.  Are  all  solutions  of  Laplace's  Equation  possible  values  of 
the  potential  function  in  empty  space  due  to  distributions  of 
matter  ?      Assume  some  particular  solution  of   this    equation 
which  will  serve  as  the  potential  function  due  to  a  possible  dis 
tribution  and  show  what  this  distribution  is. 

7.  If  the  lines  of  force  which  traverse  a  certain  region  are 
parallel,  what  may  be  inferred  about  the  intensity  of  the  force 
within  the  region  ? 

8.  The  path  of  a  material  particle  starting  from  rest  at  a 
point  P  and  moving  under  the  action  of  the  attraction  of  a  given 


72  THE    NEWTONIAN    POTENTIAL    FUNCTION 

mass  Mis  not  in  general  the  line  of  force  due  to  M which  passes 
through  P.  Discuss  this  statement,  and  consider  separately 
cases  where  the  lines  of  force  are  straight  and  where  they  are 
curved. 

9.  Draw  a  figure  corresponding  to  Figure  17  for  the  case  of 
a  uniform  sphere  of  unit  radius  surrounded  by  a  concentric 
spherical  shell  of  radii  2  and  3  respectively. 

10.  Draw  with  the  aid  of  compasses  traces  of  four  of  the 
equipotential  surfaces  due  to  two  homogeneous  infinite  cylinders 
of  equal  density  whose  axes  are  parallel  and  at  a  distance  of 
5  inches  apart,  assuming  the  radius  of  one  of  the  cylinders  to 
be  1  inch  and  that  of  the  other  to  be  2  inches. 

11.  Draw  with  the  aid  of  compasses  meridian  sections  of 
four  of  the  equipotential  surfaces  due  to  two  small  homogeneous 
spheres  of  mass  m  and  2m  respectively,  whose  centres  are  4 
inches  apart.     Can  equipotential  surfaces  be  drawn  so  as  to  lie 
wholly  or  partly  within  one  of  the  spheres?     What  value  of  the 
potential  function  gives   an  equipotential  surface   shaped   like 
the  figure  8?     Show  that  the  value  of  the  resultant  force  at  the 
point  where  this  curve  crosses  itself  is  zero. 

12.  A  sphere  of  radius  3  inches  and  of  constant  density  ^  is 
surrounded  by  a  spherical  shell  concentric  with  it  of  radii  4 
inches  and  5  inches  and  of  density  /xr,  where  r  is  the  distance 
from  the  centre.     Compute  the  values  of  the  attraction  and  of 
the  potential  function  at  all  points  in  space  and  draw  curves  to 
illustrate  the  fact  that  V  and  DrV  are  everywhere  continuous 
and  that  Dr2Fis  discontinuous  at  certain  points. 

13.  A  very  long  cylinder  of  radius  4  inches  and  of  constant 
density  ^  is  surrounded  by  a  cylindrical  shell  coaxial  with  it 
and  of  radii  6  inches  and  8  inches.     The  density  of  this  shell  is 
inversely  proportional  to  the  square  of  the  distance   from  the 
axis,  and  at  a  point  8  inches  from  this  axis  is  JJL.    Use  the  Theo 
rem  of  Gauss  to  find  the  values  of  F,  DrV,  and  Dr2V  at  differ 
ent  points  on  a  line  perpendicular  to  the  axis  of  the  cylinder  at 
its  middle  point.     If  the  value  of  the  attraction  at  a  distance 
of  20  inches  from  the  axis  is  10,  show  how  to  find  p. 


IN    THE    CASE    OF    GRAVITATION.  73 

14.  Use  Dirichlet's  value  of  DXV,  given  by  equation  [78], 
to  find  the  attraction  in  the  direction  of  the  axis  of  x  at  points 
within  a  spherical  shell  of  radii  r0  and  rx  and  of  constant  den 
sity  p. 

15.  Are  there  any  other  cases  except  those  in  which  the 
density  of  the  attracting  matter  depends  only  upon  the  dis 
tance  from  a  plane,  from  an  axis,  or  from  a  central  point, 
where  surfaces  of  equal  force  are  also  equipotential  surfaces  ? 
Prove  your  assertion. 

16.  Show  that  the  second  derivative  with  respect  to  x,  of 
the  potential  function  due  to  a  homogeneous  sphere  of  density 
p  and  radius  «,  with  centre  at  the  origin,  is  —  ^  -rrpr  for  inside 
points,  and  —  f  wpa3  (r2  —  3x2)  /  rb  for  points  without  the  sur 
face.     Similar  expressions  give  the  values  of  the  second  deriv 
atives  with  respect  to  y  and  z.     Show  that  the  normal  second 
derivative  of  V  is  —  ±  irp  just  within  the  surface  and  +  f  irp 
just  without.     Show  that  the  tangential  second  derivatives 
are  continuous  at  the  surface. 

17.  Two  uniform  straight  wires  of  length  I  and  of  masses  m^ 
and  ??i2  ai*e  parallel  to  each  other  and  perpendicular  to  the  line 
joining  their  middle  points,  which  is  of  length  y^.     Show  that 
the  amount  of  work  required  to  increase  the  distance  between 
the  wires  to  y2  by  moving  one  of  them  parallel  to  itself  is 

2m1??l2f  .-5 ,  -\Jl-  4-  u2  —  r\y=y*      r*r-  -i 

-^-b-VJ2  +  y-nog-r±-±*L.     -  I  [Mmchm.] 

L  y          Jy=yi 

18.  Show  that  if  the  earth  be  supposed  spherical  and  covered 
with  an  ocean  of  small  depth,  and  if  the  attraction  of  the  par 
ticles  of  water  on  each  other  be  neglected,  the  ellipticity  of  the 
ocean  spheroid  will  be  given  by  the  equation, 

2  _  The  centrifugal  force  at  the  equator 

g 

19.  A  spherical  shell  whose  inner  radius  is  r  contains  a  mass 
m  of  gas  which  obeys  the  Law  of  Boyle  and  Mariotte.     Find 
the  law  of  density  of  the  gas,  the  total 'normal  pressure  on  the 
inside  of  the  containing  vessel,  and  the  pressure  at  the  centre. 


74  THE    NEWTONIAN    POTENTIAL    FUNCTION 

20.  If  the  earth  were  melted  into  a  sphere  of  homogeneous 
liquid,  what  would  be  the  pressure  at  the  centre  in  tons  per 
square  foot  ?     If  this  molten  sphere  instead  of  being  homo 
geneous  had  a  surface  density  of  2.4  and  an  average  density 
of  5.6,  what  would  be  the  pressure  at  the  centre  on  the  sup 
position  that  the   density  increased  proportionately  to  the 
depth  ? 

21.  A  solid  sphere  of  attracting  matter  of  mass  ra  and  of 
radius  r  is  surrounded  by  a  given  mass  M  of  gas  which  obeys 
the  Law  of  Boyle  and  Mariotte.     If  the  whole  is  removed 
from  the  attraction  of  all  other  matter,  find  the  law  of  density 
of  the  gas  and  the  pressure  on  the  outside  of  the  sphere. 

22.  The  potential  function  within  a  closed  surface  S  due  to 
matter  wholly  outside  the  surface  has  for  extreme  values  the 
extreme  values  upon  S. 

23.  If  the  potential  functions  V  and  V  due  to  two  systems 
of  matter  without  a  closed  surface  have  the  same  values  at  all 
points  on  the  surface,  they  will  be  equal  throughout  the  space 
enclosed  by  the  surface. 

24.  The  potential  function  outside  of  a  closed  surface  due 
to  matter  wholly  within  the  surface  has  for  its  extreme  values 
two  of  the  following  three  quantities  :    zero  and  the  extreme 
values  upon  the  surface. 

[Answers  to  some  of  these  problems  and  a  collection  of  additional  prob 
lems  illustrative  of  the  text  of  this  chapter  may  be  found  near  the  end  of 
the  book.] 


IN    THE    CASE    OF    REPULSION.  75 


CHAPTER   III. 

THE  POTENTIAL  FUNCTION  IN  THE  CASE  OP 
EEPULSION, 

39.  Repulsion,  according  to  the  Law  of  Nature.  Certain 
physical  phenomena  teach  us  that  bodies  ma}'  acquire,  by 
electrification  or  otherwise,  the  property  of  repelling  each  other, 
and  that  the  resulting  force  of  repulsion  between  two  bodies  is 
often  much  greater  than  the  force  of  attraction  which,  ac 
cording  to  the  Law  of  Gravitation,  every  body  has  for  every 
other  bod}7. 

Experiment  shows  that  almost  every  such  case  of  repulsion, 
however  it  may  be  explained  physically,  can  be  quantitatively 
accounted  for  by  assuming  the  existence  of  some  distribution  of 
a  kind  of"  matter,"  every  particle  of  which  is  supposed  to  repel 
every  other  particle  of  the  same  sort  according  to  the  "  Law  of 
Nature,"  that  is,  roughly  stated,  with  a  force  directly  propor 
tional  to  the  product  of  the  quantities  of  matter  in  the  particles, 
and  inversely  proportional  to  the  square  of  the  distance  between 
their  centres. 

In  this  chapter  we  shall  assume,  for  the  sake  of  argument, 
that  such  matter  exists,  and  proceed  to  discuss  the  effects  of 
different  distributions  of  it.  Since  the  law  of  repulsion  which 
we  have  assumed  is,  with  the  exception  of  the  opposite  direc 
tions  of  the  forces,  mathematically  identical  with  the  law  which 
governs  the  attraction  of  gravitation  between  .particles  of  pon 
derable  matter,  we  shall  find  that,  bjr  the  occasional  intro 
duction  of  a  change  of  sign,  all  the  formulas  which  we  have 
proved  to  be  true  for  cases  of  attraction  due  to  gravitation 
can  be  made  useful  in  treating  corresponding  problems  in 
repulsion. 


T6 


THE    POTENTIAL,    FUNCTION 


40.  Force  at  Any  Point  due  to  a  Given  Distribution  of 
Repelling  Matter.  Two  equal  quantities  of  repelling  matter 
concentrated  at  points  at  the  unit  distance  apart  are  called 
"  unit  quantities"  when  they  are  such  as  to  make  the  force  of 
repulsion  between  them  the  unit  force. 

If  the  ratio  of  the  quantity  of  repelling  matter  within  a  small 
closed  surface  supposed  drawn  about  a  point  P,  to  the  volume 
of  the  space  enclosed  by  the  surface,  approaches  the  limit  p  when 
the  surface  (always  enclosing  P)  is  supposed  to  be  made  smaller 
and  smaller,  p  is  called  the  "density"  of  the  repelling  matter 
at  P. 

In  order  to  find  the  magnitude  at  any  point  P  of  the  force  due 
to  any  given  distribution  of  repelling  matter,  we  may  suppose 
the  space  occupied  by  this  matter  to  be  divided  up  into  small 
elements,  and  compute  an  approximate  value  of  this  force  on  the 
assumption  that  each  element  repels  a  unit  quantity  of  matter 
concentrated  at  P  with  a  force  equal  to  the  quantity  of  matter 
in  the  element  divided  by  the  square  of  the  distance  between  P 
and  one  of  the  points  of  the  element.  The  limit  approached  by 
this  approximate  value  as  the  size  of  the  elements  is  diminished 
indefinitely  is  the  value  required. 


FIG. 


Let  Q  (Fig.  30),  whose  coordinates  are  a;',  ?/r,  z',  be  the 
corner  next  the  origin  of  an  element  of  the  distribution.  Let  p 
be  the  density  at  Q  and  Aa;'A//Az'  the  volume  of  the  element; 
then  the  force  at  P  due  to  the  matter  in  the  element  is  approxi- 


IN    THE    CASE    OF    REPULSION.  77 

mately  equivalent  to  a  force  of  magnitude  p    X     ^ — —  acting  in 
the  direction  QP,  or  a  force  of  magnitude  —  £ —       ^  ._  acting 

]_)/  \~ 

in  the  direction  PQ.     If  the  coordinates  of  P  are  #,  y,  z,  the 
component  of  this  force  in  the  direction  of  the  positive  axis  of  x 


is  aud  thc  forcc  atP          llel 

'22' 


to  the   axis  of  x  due   to  the   whole  distribution   of   repelling 
matter  is 


\-=  -  -  ri.)5  -, 

JJJ[(.l-'-x)2+(2/'-</)2+(2'-^)2]i' 

where  the  triple  integration  is  to  be  extended  over  the  whole 
space  filled  with  the  repelling  matter.  For  the  components  of 
the  force  at  P  parallel  to  the  other  axes  we  have,  similarly, 


and 


-      C  C  C  P(y'-y)dx'dy'dz'  ri9«M 

J  J  J  [(*'-aO«+(y'-jO*+(*'-z)»]r 

=  -CCC P(z'-z)dMyW ri251 

J  J  J  [(.v-.x')2+(y-2/)2+(^-^)^j^ 


If  we  denote  by  V  the  function 

pdx'dy'dz1 


OT*'         [126] 

which,  together  with  its  first  derivatives,  is  everywhere  finite 
and  continuous,  as  we  have  shown  in  the  last  chapter,  it  is  easy 
to  see  that 

X=-DXV,     Y=-DyV,     Z  =  -DZV,  [127] 

T2,  [128] 


and  that  the  direction-cosines  of  the  line  of  action  of  the  re 
sultant  force  at  P  are 


(  O  THE    POTENTIAL    FUNCTION 

It  follows  from  this  (see  Section  21)  that  the  component  in 
any  direction  of  the  force  at  a  point  P  due  to  any  distribution 
M  of  repelling  matter  is  minus  the  value  at  P  of  the  partial 
derivative  of  the  function  V  taken  in  that  direction. 

The  function  Fgoes  by  the  name  of  the  Newtonian  potential 
function  whether  we  are  dealing  with  attracting  or  repelling 
matter. 

In  the  case  of  repelling  matter,  it  is  evident  that  the  resultant 
force  on  a  particle  of  the  matter  at  any  point  tends  to  drive  that 
particle  in  a  direction  which  leads  to  points  at  which  the  poten 
tial  function  has  a  lower  value,  whereas  in  the  case  of  gravita 
tion  a  particle  of  ponderable  matter  at  any  point  tends  to  move 
in  a  direction  along  which  the  potential  function  increases. 

41.  The  Potential  Function  as  a  Measure  of  Work.     It  is 

easy  to  show  by  a  method  like  that  of  Article  27  that  the 
amount  of  work  required  to  move  a  unit  quantity  of  repelling 
matter,  concentrated  at  a  point,  from  Px  to  P2,  in  face  of  the 
force  due  to  any  distribution  M  of  the  same  kind  of  matter,  is 
F2  —  VH  where  FI  and  F2  are  the  values  at  Px  and  P2  respec 
tively  of  the  potential  function  due  to  M.  The  farther  P1  is 
from  the  given  distribution,  the  smaller  is  Pi,  and  the  less  does 
F2  —  FI  differ  from  F2.  In  fact,  the  value  of  the  potential 
function  at  the  point  P2,  wherever  it  may  be,  measures  the  work 
which  would  be  required  to  move  the  unit  quantity  of  matter  by 
any  path  from  "  infinity"  to  P2. 

42.  Gauss's  Theorem  in  the  Case  of  Repelling  Matter.     If  a 
quantity  m  of  repelling  matter  is  concentrated  at  a  point  within 
a  closed  oval  surface,  the  resultant  force  due  to  m  at  any  point 
on  the  surface  acts  toward  the  outside  of  the  surface  instead  of 
towards  the  inside,  as  in  the  case  of  attracting  matter. 

Keeping  this  in  mind,  we  may  repeat  the  reasoning  of  Article 
31,  using  repelling  matter  instead  of  attracting  matter,  and  sub 
stituting  all  through  the  work  the  exterior  normal  for  the  in 
terior  normal,  and  in  this  way  prove  that : 


IN    THE    CASE    OF    REPULSION.  79 

If  there  be  any  distribution  of  repelling  matter  partly  within 
and  partly  without  a  closed  surface  T,  and  if  M  be  the  whole 
quantity  of  this  matter  enclosed  by  T7,  and  M1  the  quantity  out 
side  T,  the  surface  integral  over  T  of  the  component  in  the  di 
rection  of  the  exterior  normal  of  the  force  due  to  both  M  and  M' 
is  equal  to  4  irM.  If  V  be  the  potential  function  due  to  M  and 
W,  we  have 


43.  Poisson's  Equation  in  the  Case  of  Repelling  Matter.  If 
we  applv  the  theorem  of  the  last  article  to  the  surface  of  a 
volume  element  cut  out  of  space  containing  repelling  matter, 
and  use  the  notation  of  Article  35,  we  shall  find  that  in  the  case 
of  rectangular  coordinates  the  surface  integral,  taken  over  the 
element,  of  the  component  in  the  direction  of  the  exterior 
normal  is 

A*  Ay  A*  +        -  +  =  4  irft,  •  A*  Ay  A*,      [130] 


where  X  is  the  average  component  in  the  positive  direction  of 
the  axis  of  x  of  the  force  on  the  elementary  surface  A?/Az,  and 
where  T  and  Z  have  similar  meanings.  It  is  evident  that  if 
the  element  be  made  smaller  and  smaller,  X,  1%  and  Z  will 
approach  as  limits  the  components  parallel  to  the  coordinate 
axes  of  the  force  at  P.  These  components  are  —  DXV,  —  DyV. 
and  —  DZV',  so  that  if  we  divide  [130]  by  A#A?/Az  and  then 
decrease  indefinitely  the  dimensions  of  the  element,  we  shall 
arrive  at  the  equation 

V2F=-47rp.  [131] 

B}*  using  successively  cylinder  coordinates  and  spherical  co 
ordinates  we  may  prove  the  equations 


=  -  47rp,  [132] 

and          sin0  -  D,(i*DrV)  +  -  +  D0(sin6  -  D9V) 

[133] 


80  THE    POTENTIAL    FUNCTION 

so  that  Poisson's  Equation  holds  whether  we  are  dealing  will: 
attracting  or  repelling  matter. 

44.  Coexistence  of  Two  Kinds  of  Active  Matter.  Certain 
physical  phenomena  may  be  most  conveniently  treated  mathe 
matically  by  assuming  the  coexistence  of  two  kinds  of  "matter" 
such  that  any  quantity  of  either  kind  repels  all  other  matter  of 
the  same  kind  according  to  the  Law  of  Nature,  and  attracts  all 
matter  of  the  other  kind  according  to  the  same  law. 

Two  quantities  of  such  matter  may  be  considered  equal  if, 
when  placed  in  the  same  position  in  a  field  of  force,  they  are 
subjected  to  resultant  forces  which  are  equal  in  intensity  and 
which  have  the  same  line  of  action.  The  two  quantities  of 
matter  are  of  the  same  kind  if  the  direction  of  the  resultant 
forces  is  the  same  in  the  two  cases,  but  of  different  kinds  if  the 
directions  are  opposed.  The  unit  quantity*  of  matter  is  that 
quantity  which  concentrated  at  a  point  would  repel  with  the 
unit  force  an  equal  quantity  of  the  same  kind  concentrated  at 
a  point  at  the  unit  distance  from  the  first  point. 

It  is  evident  from  Articles  2,  14,  and  40  that  m  units  of  one 
of  these  kinds  of  matter,  if  concentrated  at  a  point  (#,  ?/,  z)  and 
exposed  to  the  action  of  m1?  w2,  m^  ...  mk  units  of  the  same 
kind  of  matter  concentrated  respectively  at  the  points  (o^,  2/1,^1), 
(Eg,  ?/2,  z2),  Ov  ?/.„  z3),  ...  Ov  ?/,,  ZA),  and  of  mk+l,  mk+2,  ...  mn 
units  of  the  other  kind  of  matter  concentrated  respectively  at 
the  points  (xk  +  l,  ykJt  1:  zA  +  1),  (ajM.2,  yk  +  2,  zk  +  2),  ...  (xn,  yn,  ZH), 
will  be  urged  in  the  direction  parallel  to  the  positive  axis  of  x 
with  the  force 

fe^,         [134] 


i=l  t=*+l 

where   rt   is   the    distance   between  the   points    (#,   y,  z)    and 


*  With  this  definition  of  the  unit  of  quantity,  the  repulsion  and  attrac 
tion  force  unit  is  identical  with  the  absolute  kinetic  force  unit. 


IN    THE    CASE    OF    REPULSION  81 

If  we  agree  to  distinguish  the  two  kinds  of  matter  from  each 
other  by  calling  one  kind  "  positive  "  and  the  other  kind  "  neg 
ative,"  it  is  easy  to  see  that  if  every  m  which  belongs  to  positive 
matter  be  given  the  plus  sign  and  every  m  which  belongs  to 
negative  matter  the  minus  sign,  we  may  write  the  last  equation 
in  the  form 

-=^-  [135] 


The  result  obtained  by  making  m  in  [135]  equal  to  unity  is 
called  the  force  at  the  point  (x,  y,  z). 

In  general,  m  units  of  either  kind  of  matter  concentrated  at 
the  point  (#,  y,  z)  ,  and  exposed  to  the  action  of  any  continuous 
distribution  of  matter,  will  be  urged  in  the  positive  direction  of 
the  axis  of  x  b  the  force 


iii  this  expression,  p,  the  density  at  (x',  ?/',  z') ,  is  to  be  taken 
positive  or  negative  according  as  the  matter  at  the  point  is 
positive  or  negative  :  m  is  to  have  the  sign  belonging  to  the 
matter  at  the  point  (x,  y,  z)  :  and  the  limits  of  integration  are  to 
be  chosen  so  as  to  include  all  the  matter  which  acts  on  m. 

With  the  same  understanding  about  the  signs  of  m  and  of  p, 
it  is  clear  that  the  force  which  urges  in  any  direction  s,  m  units 
of  matter  concentrated  at  the  point  (#,?/,  z)  is  equal  to  — m-D4F, 
where  F'is  the  everywhere  finite,  continuous,  and  single-valued 

function 

pdx'di/'dz' 


///IB= 


and  that  mV measures  the  amount  of  work  required  to  bring  up 
from  u  infinity"  by  any  path  to  its  present  position  the  m  units 
of  matter  now  at  the  point  (#,  ?/,  2) . 

If  we  call  the  resultant  force  which  would  act  on  a  unit  of 
positive  matter  concentrated  at  the  point  P  "-the  force  at  P" 


82  THE    POTENTIAL    FUNCTION. 

it  is  clear  that  if  any  closed  surface  T  be  drawn  in  a  field  of 
force  due  to  any  distribution  of  positive  and  negative  matter  so 
as  to  include  a  quantity  of  this  matter  algebraically  equal  to  Q, 
the  surface  integral  taken  over  T  of  the  component  in  the  direc 
tion  of  the  exterior  normal  of  the  force  at  the  different  points  of 
the  surface  is  equal  to  4=irQ. 

It  will  be  found,  indeed,  that  all  the  equations  and  theorems 
given  earlier  in  this  chapter  for  the  case  of  one  kind  of  repelling 
matter  may  be  used  unchanged  for  the  case  where  positive  and 
negative  matter  coexist,  if  we  only  give  to  p  and  m  their  proper 
signs. 

It  is  to  be  noticed  that  Poisson's  Equation  is  applicable 
whether  we  are  dealing  with  attracting  matter  or  repelling  mat 
ter,  or  positive  and  negative  matter  existing  together. 

EXAMPLES. 

1.  Show  that  the  extreme  values  of  the  potential  function 
outside  a  closed  surface  S,  due  to  a  quantity  of  matter  algebrai 
cally  equal  to  zero  within  the  surface,  are  its  extreme  values 
on£. 

2.  Show  that  if  the  potential  function  due  to  a  quantity  of 
matter  algebraically  equal  to  zero  and  shut  in  by  a  closed  sur 
face  S  has  a  constant  value  all  over  the  surface,  then  this  con 
stant  value  must  be  zero. 


GREEN  S   THEOREM. 


83 


CHAPTER    IV. 

SUKFAOE  DISTRIBUTIONS. -GREEK'S  THEOREM, 

45.  Force  due  to  a  Closed  Shell  of  Repelling  Matter.  If  a 
quantity  of  very  finely-divided  repelling  matter  be  enclosed  in  a 
box  of  any  shape  made  of  indifferent  material,  it  is  evident 
from  [127]  and  from  the  principles  of  Section  38  that  if  the  vol 
ume  of  the  box  is  greater  than  the  space  occupied  by  the  repel 
ling  matter,  the  latter  will  arrange  itself  so  that  its  free  surface 
will  be  equipotential  with  regard  to  all  the  active  matter  in 
existence,  taking  into  account  any  there  may  be  outside  the  box 
as  well  as  that  inside.  It  is  easy  to  see,  moreover,  that  we 
shall  have  a  shell  of  matter  lining  the  box  and  enclosing  an 
empty  space  in  the  middle. 

That  any  such  distribution  as  that  indicated  in  the  subjoined 
diagram  is  impossible  follows  immediately  from  the  reasoning 
of  Section  37.  For  ABC  and  DEF  are  parts  of  the  same 


equipotential  free  surface  of  the  matter.  If  we  complete  this 
surface  by  the  parts  indicated  by  the  dotted  lines,  we  shall 
enclose  a  space  void  of  matter  and  having  therefore  throughout 
a  value  of  the  potential  function  equal  to  that  on  the  bounding 


84  SURFACE    DISTRIBUTIONS. 

surface.  But  in  this  case  all  points  which  can  be  reached  from 
0  by  paths  which  do  not  cut  the  repelling  matter  must  be  at  the 
same  potential  as  0,  and  this  evidently  includes  all  space  not 
actually  occupied  by  the  repelling  matter ;  which  is  absurd. 

Let  us  consider,  then  (see  Fig.  82),  a  closed  shell  of  repelling 
matter  whose  inner  surface  is  equi potential,  so  that  at  every 
point  of  the  cavity  which  the  shell  shuts  in,  the  resultant  force, 
due  to  the  matter  of  which  the  shell  is  composed  and  to  any 
outside  matter  there  may  be,  is  zero. 

Let  us  take  a  small  portion  w  of  the  bounding  surface  of  the 
cavity  as  the  base  of  a  tube  of  force  which  shall  intercept  an 


FIG.  32. 

area  <o'  on  an  equipotential  surface  which  cuts  it  just  outside  the 
outer  surface  of  the  shell,  and  let  us  apply  Gauss's  Theorem  to 
the  box  enclosed  by  w,  <o',  and  the  tube  of  force.  If  F'  is  the 
average  value  of  the  resultant  force  on  o>',  the  only  part  of  the 
surface  of  the  box  which  yields  anything  to  the  surface  integral 
of  normal  force,  we  have 

F'u'  =  4  urn, 

where  m  is  the  quantity  of  matter  within  the  box.  If  we  multi 
ply  and  divide  by  o>,  this  equation  may  be  written 

F,=  4*mm»_t  [137] 

to  0> 

If  to  be  made  smaller  and  smaller,  so  as  alwa}Ts  to  include  a 
given  point  A,  to1  as  it  approaches  zero  will  always  include  a 
point  B  on  the  line  of  force  drawn  through  ^4,  and  F'  will  ap 
proach  the  value  F  of  the  resultant  force  at  B. 

The  shell  may  be  regarded  as  a  thick  layer  spread  upon  the 


GREEN'S  THEOREM.  85 

inner  surface,  and  in  this  case  the  limit  of  --  may  be  consid 
er 

ered  the  value  at  A  of  the  rate  at  which  the  matter  is  spread 
upon  the  surface.     If  we  denote  this  limit  by  <r,  we  shall  have 


If  B  be  taken  just  outside  the  shell,  and  if  the  latter  be  very 
thin,  ^^Q  f~]  evidently  differs  little  from  unity;  and  we  see 

that  the  resultant  force  at  a  point  just  outside  the  outer  sur 
face  of  a  shell  of  matter,  whose  inner  surface  is  equipotential, 
becomes  more  and  more  nearly  equal  to  4  TT  times  the  quantity 
of  matter  per  unit  of  surface  in  the  distribution  at  that  point  as 
the  shell  becomes  thinner  and  thinner. 

The  reader  may  find  out  for  himself,  if  he  pleases,  whether  or 
not  the  line  of  action  of  the  resultant  force  at  a  point  just  out 
side  such  a  shell  as  we  have  been  considering  is  normal  to  the 
shell. 

It  is  to  be  carefully  noticed  that  the  inner  surface  of  a  closed 
shell  need  not  be  equipotential  unless  the  matter  composing  the 
shell  is  finely  divided  and  free  to  arrange  itself  at  will. 

"When  the  shell  is  thin$  and  we  regard  it  as  formed  of  matter 
spread  upon  its  inner  surface,  <r  is  called  the  "surface  density  " 
of  the  distribution,  and  its  value  at  any  point  of  the  inner  sur 
face  of  the  shell  may  be  regarded  as  a  measure  of  the  amount  of 
matter  which  must  be  spread  upon  a  unit  of  surface  if  it  is  to 
be  uniformly  covered  with  a  layer  of  thickness  equal  to  that  of 
the  shell  at  the  point  in  question. 

46.  Surface  Distributions.  It  often  becomes  necessary  in  the 
mathematical  treatment  of  physical  problems,  on  the  assump 
tion  of  the  existence  of  a  kind  of  repelling  matter  or  agent,  to 
imagine  a  finite  quantity  of  this  agent  condensed  on  a  surface 
in  a  layer  so  thin  that  for  practical  purposes  we  may  leave  the 
thickness  out  of  account.  If  a  shell  like  that  considered  in  the 
last  section  could  be  made  thinner  and  thinner  by  compression 


86 


SURFACE    DISTRIBUTIONS. 


while  the  quantity  of  matter  in  it  remained  unchanged,  the 
volume  density  (p)  of  the  shell  would  grow  larger  and  larger 
without  limit,  and  a  would  remain  finite.  A  distribution  like 
this,  which  is  considered  to  have  no  thickness,  is  called  a  sur 
face  distribution. 

The  value  at  a  point  .£*  of  the  potential  function  due  to 
a  superficial  distribution  of  surface  density  a-  is  the  surface 
integral,  taken  over  the  distribution,  of  -,  where  r  is  the  dis 
tance  from  P. 

It  is  evident  that  as  long  as  P  does  not  lie  exactly  in  the 
distribution,  the  potential  function  and  its  derivatives  are  alwa}Ts 
finite  and  continuous,  and  the  force  at  any  point  in  any  direc 
tion  may  be  found  by  differentiating  the  potential  function 
partially  with  regard  to  that  direction. 

If  p  were  infinite,  the  reasoning  of  Article  22  would  no 
longer  apply  to  points  actually  in  the  active  matter,  and  it  is 
worth  our  while  to  prove  that  in  the  case  of  a  surface  distri 
bution  where  a-  is  everywhere  finite,  the  value  at  a  point  P  of 
the  potential  function  due  to  the  distribution  remains  finite,  as 


FIG.  33. 

P  is  made  to  move  normally  through  the  surface  at  a  point  of 
finite  curvature. 

To  show  this,  take  the  point  0  (Fig.  33),  where  P  is  to  cut 
the  surface,  as  origin,  and  the  normal  to  the  surface  at  0  as 


GKEEN'S  THEOREM.  87 

the  axis  of  £,  so  that  the  other  coordinate  axes  shall  lie  iu 
the  tangent  plane. 

If  the  curvature  in  the  neighborhood  of  0  is  finite,  it  will  be 
possible  to  draw  on  the  surface  about  0  a  closed  line  such  that 
for  every  point  of  the  surface  within  this  line  the  normal  will 
make  an  acute  angle  with  the  axis  of  x. 

For  convenience  we  will  draw  the  closed  line  of  such  a  shape 
that  its  projection  on  the  tangent  plane  shall  be  a  circle  whose 
centre  is  at  0  and  whose  radius  is  £7,  and  we  will  cut  the  area 
shut  in  by  this  line  into  elements  of  such  shape  that  their  pro 
jections  upon  the  tangent  plane  shall  divide  the  circle  just 
mentioned  into  elements  bounded  by  concentric  circumferences 
drawn  at  radial  intervals  of  A«,  and  by  radii  drawn  at  angular 
distances  of  A<£. 

If  a;,  0,  0  are  the  coordinates  of  the  point  P,  a;',  y\  z*  those 
of  a  point  of  one  of  the  elements  of  the  area  shut  in  by  the 
closed  line,  and  a  the  angle  which  the  normal  to  the  surface 
at  this  point  makes  with  the  axis  of  x,  the  size  of  the  surface 

element  is  approximately  -  —  -  —  —  ,  where  u2  =  z'2-\-  ?/'2,  and  the 

COSa 

value  at  P  of  the  potential  function  due  to  that  part  of  the  sur 
face  distribution  shut  in  by  the  closed  line  is 


The  quantity 

all  or  sec  a 


cos  a  V(a  —  x')2  -}-  u2 


is  always  finite,  for,  whatever  the  value  of  the  quantity  under 

the  radical  sign  in  the  last  expression  may  be  when  x.  #',  and  " 

are  all  zero,  it  cannot  be  less  than  unity,  and  therefore  Vi  must       1  JL>i0  $ 

be  finite  even  when  P  moves  down  the  axis  of  x  to  the  surface 

itself. 

If  V  and  F2  are  the  values  at  P  of  the  potential  functions 
due  respectively  to  all  the  existing  acting  matter  and  to  that 


88  SURFACE    DISTRIBUTIONS. 

part  of  this  matter  not  lying  on  the  portion  of  the  surface  shut 
in  by  our  closed  line,  we  have  V=Vl  +  V%,  and,  since  P  is  a 
point  outside  the  matter  which  gives  rise  to  V^  the  latter  is 
finite  ;  so  that  Fis  finite. 

The  reader  who  wishes  to  study  the  properties  of  the  deriva 
tives  of  the  potential  function,  and  their  relations  to  the  force 
components  at  points  actually  in  a  surface  distribution,  will  find 
the  whole  subject  treated  in  the  first  part  of  Riemann's  Schwere, 
Electricitat  und  Magnetismus. 

Using  the  notation  of  this  section,  it  is  easy  to  write  down 
definite  integrals  which  represent  the  values  of  the  potential 
function  at  two  points  on  the  same  normal,  one  on  one  side  of 
a  superficial  distribution,  and  at  a  distance  a  from  it,  and  the 
other  on  the  other  side  at  a  like  distance,  and  to  show  that  the 
difference  between  these  integrals  may  be  made  as  small  as  we 
like  by  choosing  a  small  enough.  This  shows  that  the  value  of 
the  potential  function  at  a  point  P  changes  continuously,  as  P 
moyes  normally  through  a  surface  distribution  of  finite  super 
ficial  density.  If  matter  could  be  concentrated  upon  a  geo 
metric  line,  so  that  there  should  be  a  finite- quantity  of  matter 
on  the  unit  of  length  of  the  line,  or  if  a  finite  quantity  of  matter 
could  be  really  concentrated  at  a  point,  the  resulting  potential 
function  would  be  infinite  on  the  line  itself,  and  at  the  point. 

47.  The  Normal  Force  at  Any  Point  of  a  Surface  Distribu 
tion.  In  the  case  of  a  strictly  superficial  distribution  on  a 
closed  surface  where  the  repelling  matter  is  free  to  arrange 
itself  at  will,  the  inner  surface  of  the  matter  (and  hence  the 
outer  surface,  which  is  coincident  with  it)  is  equipotential,  and 
the  resultant  force  at  a  point  B  just  outside  the  distribution  is 
normal  to  the  surface  and  numerically  equal  to  4?r  times  the 
surface  density  at  23.  This  shows  that  the  derivative  of  the 
potential  function  in  the  direction  of  the  normal  to  the  surface 
has  values  on  opposite  sides  of  the  surface  differing  by  4  TTO-, 
and  at  the  surface  itself  cannot  be  said  to  have  any  definite 
value. 


GREEN'S  THEOREM.  89 

It  is  easy,  however,  to  find  the  force  with  which  the  repelling 
matter  composing  a  superficial  distribution  is  urged  outwards. 
For,  take  a  small  element  o>  of  the  surface  as  the  base  of  a  tube 
of  force,  and  apply  Gauss's  Theorem  to  a  box  shut  in  by  the 
surface  of  distribution,  the  tube  of  force,  and  a  portion  o>'  of 
an  equipoteutial  surface  drawn  just  outside  the  distribution. 
Let  F  and  F'  be  the  average  forces  at  the  points  of  o>  and  <o' 
respectively,  then  the  surface  integral  of  normal  forces  taken 
over  the  box  is  F'u'  —  Fa).  and  this,  since  the  only  active 
matter  is  concentrated  on  the  surface  of  the  box  (see  Section 
31),  is  equal  to  27ro-0(o,  where  o-0  is  the  average  surface  density 
at  the  points  of  the  element  o>.  This  gives  us 

F=F'  —  —  2 


7TO-,,. 
O) 


Now  let  the  equipotential  surface  of  which  co'  is  a  part  be 
drawn  nearer  and  nearer  the  distribution  ;  then 

lim—  =  1,     lim  F'  =  47ro-0,     and    F=  27r<r0. 

F  is  the  average  force  which  would  tend  to  move  a  unit  quan 
tity  of  repelling  matter  concentrated  successively  at  the  differ 
ent  points  of  co  in  the  direction  of  the  exterior  normal,  but  the 
actual  distribution  on  co  is  COCTO,  so  that  this  matter  presses  on 
the  medium  which  prevents  it  from  escaping  with  the  force 
27Tcr02co;  and,  in  general,  the  pressure  exerted  on  the  resisting 
medium  which  surrounds  a  surface  distribution  of  repelling 
matter  is  at  am*  point  2 TTOT  per  unit  of  surface,  where  or  is  the 
surface  density  of  the  distribution  at  the  point  in  question. 

We  may  imagine  a  super licial  distribution  of  matter  which  is 
fixed,  instead  of  being  free  to  arrange  itself  at  will.  In  this 
case  the  surface  of  the  matter  will  not  be  in  general  equipoten 
tial,  but,  if  we  apply  Gauss's  Theorem  to  a  box  shut  in  bv  a 
slender  tube  of  force  traversing  the  distribution,  and  by  two 
surfaces  drawn  parallel  to  the  distribution  and  close  to  it,  one 
on  one  side  and  one  on  the  other,  we  may  prove  that  the 


90  SURFACE  'DISTRIBUTIONS. 

normal  component  of  the  force  at  a  point  just  outside  the  dis 
tribution  differs  by  4  TTO-  from  the  normal  component,  in  the 
same  sense,  of  the  force  at  a  point  just  inside  the  distribution 
on  the  line  of  force  which  passes  through  the  first  point. 

It  is  sometimes  convenient  to  denote  the  "charge"  on  a 
small  area  about  a  point  P  on  a  surface  distribution  by  A', 
and  the  rest  of  the  distribution  by  A",  and  to  consider  sepa 
rately  the  effects  of  A'  and  A".  If  Pl  and  P2  are  points  on 
the  normal  to  the  surface  drawn  through  P  and  near  the 
surface  on  opposite  sides  of  it ;  if  NI,  N^'  are  the  components 
in  the  direction  PPl  of  the  forces  at  P^  due  to  A'  and  A" 
respectively,  and  if  Nz',  N2"  are  the  corresponding  components 
at  Pz  in  the  direction  PPZ,  then  if  Pl  and  Pz  approach  P, 

lim  [-ZV/  +  W  +  NJ  +  ^V2"]  -  4  TTO-, 

where  a-  is  the  density  of  the  distribution  at  P.  The  force 
due  to  A"  changes  continuously  as  P^  moves  toward  P2,  however 
small  A'  may  be,  so  that 

lim  N,"  =  -  lim  N9"  and  lim  (NJ  +  N2')  =  4  TTO-, 

and,  by  choosing  A'  small  enough,  we  may  make  N~i  to  differ 
in  numerical  value  as  little  as  we  please  from  lim  N2'  or  from 

27TO-. 

If  the  surface  distribution  is  equipotential,  and  if  it  shuts 
in  a  region  of  no  force,  then  if  Pl  is  in  this  region,  JV~/ =  —  JV/', 
so  that  JVi"  and  N2"  can  be  made  to  differ  as  little  as  one 
pleases  in  numerical  value  from  2  no-  by  making  A'  small 
enough.  Let  the  element  of  area  covered  by  A'  be  <o  and  the 
surface  density  of  the  charge  on  it  o-,  then  the  force  with 
which  A'  is  urged  in  a  direction  normal  to  the  surface  by  A" 
is  coo-  •  2  TTO-  within  an  infinitesimal  of  higher  order  than  <o. 
That  is,  whatever  the  sign  of  o-,  the  surface  distribution  may 
be  said  to  urge  the  surrounding  medium  outwards  with  a 
pressure  in  force  units  per  unit  of  area  which  at  P  has  the 
value  2  Tro-2,  as  we  have  already  seen. 

It  is  easy  to  show  that  even  if  the  surface  distribution  is 
not  equipotential  the  components  at  Pl  and  P2  of  the  force 


GREEN  S    THEOREM. 


91 


in  any  fixed  direction   parallel  to  the  surface  approach  the 
same  limit  as  Pl  and  P2  approach  P. 

At  any  point  P  of  an  equipotential  surface  covered  with  a 
superficial  distribution  of  density  o-  the  normal  second  deriv 
ative  of  V  has  a  discontinuity*  of  4  TTO-  (  —  +  —  )  where  El 

\J*l       MZ/ 

and  R%  are  the  radii  of  curvature  at  P  of  two  mutually  per 
pendicular  normal  sections  of  the  equipotential  surface. 

48.  Green's  Theorem,  Before  proving  a  very  general  theo 
rem  due  to  Green,  t  of  which  what  we  have  called  Gauss's 
Theorem  is  a  special  case,  we  will  show  that  if  S  is  any  closed 
surface  and  U  a  function  of  a*,  y,  and  z,  which  for  every  point 
inside  S  is  continuous,  and  single-valued, 

f  C  Cl)x U •  dx dy d~  =  Cu-  J)nx  •  ds,  [140] 

where  the  first  integral  is  to  include  all  the  space  shut  in  by 
S,  and  the  second  is  to 
be  taken  over  the  whole 
surface,  and  where  Dnoc 
represents  the  deriva 
tive  of  x  taken  in  the 
direction  of  the  exte 
rior  normal. 

To  prove  this,  choose 
the  coordinate  axes  so 
that  S  shall  lie  in  the 
first  octant,  and  divide 
the  space  inside  the 
contour  of  the  projection  of  S  on  the  plane  yz  into  elements  of 
size  dydz.  On  each  of  these  elements  erect  a  right  prism  cutting 
S  twice  or  some  other  even  number  of  times.  Let  us  call  the 
values  of  U  at  the  successive  points  where  the  edge  nearest  the 

*  C.  Neumann,  Math.  Ann.  1880.  Th.  Horn,  Zeitschr.  f.  Math.  u. 
Phys.  1881. 

t  George  Green,  An  Essay  on  the  Application  of  Mathematical  Analy 
sis  to  the  Theories  of  Electricity  and  Magnetism.  Nottingham,  1828. 


92  SURFACE    DISTRIBUTIONS. 

axis  of  x  of  any  one  of  these  prisms  cuts  S  ;  Ul}  U2,  £73,  •  •  •  U2n 
respectively  ;  the  angles  which  this  edge  makes  with  exterior 
normals  drawn  to  S  at  these  points,  a1?  a2,  a3,  •  •  •  a2w  ;  and  the 
elements  which  the  prism  cuts  from  the  surface  S  ;  ds^  ds2, 
dss,  •  •  -ds,2n.  It  is  evident  that  wherever  a  line  perpendicular  to 
the  plane  yz  cuts  into  S,  the  corresponding  value  of  a  is  obtuse 
and  its  cosine  negative,  but  wherever  such  a  line  cuts  out  of  St 
the  corresponding  value  of  a  is  acute  and  its  cosine  positive. 

Keeping  this  iu  mind,  we  shall  see  that  although  the  base  of 
a  prism  is  the  common  projection  of  all  the  elements  which  it 
cuts  fro'm  $,  and  in  absolute  value  is  approximately  equal  to 
any  one  of  these  multiplied  by  the  corresponding  value  of  cos  a, 
yet,  since  dxdy,  ds^  c?s2,  etc.,  are  all  positive  areas  and  some  of 
the  cosines  are  negative,  we  must  write,  if  we  take  account  of  signs, 


dydz  =  —  dsjcosci!  =  +^2cosa2  =  —  ds3cosa3  =  •••„ 


If  the  indicated  integration  with  regard  to  x  in  the  left-hand 
member  of  [140]  be  performed  and  the  proper  limits  introduced, 
we  shall  have 

'  U,+  £74-...],  [141] 

where  the  double  sign  of  integration  directs  us  to  form  a  quan 
tity  corresponding  to  that  in  brackets  for  every  prism  which 
cuts  $,  to  multiply  this  by  the  area  of  the  base  of  the  prism, 
and  to  find  the  limit  of  the  sum  of  all  the  results  as  the  bases  of 
the  prisms  are  made  smaller  and  smaller. 

Since  we  may  substitute  for  dydz  any  one  of  its  approxi 
mate  values  given  above,  we  may  write  the  quantity  within 
the  brackets 

U\  COS  ttj  ds1  +  U2  COS  a  2  ds2  +  U3  COS  a;i  (7.S.,  -J-  •  •  •  , 

and  this  shows  that  the  double  integral  is  equivalent  to  the  sur 
face  integral,  taken  over  the  whole  of  $,  of  £7  cos  a,  whence  we 
may  write 

C  C  Cl>,U-da;dyciz=  Cuco&ada,  [1-42] 


GREEN'S  THEOREM.  93 

where  the  first  integral  is  to  be  taken  all  through  the  space 
shut  in  by  S,  and  the  second  over  the  whole  surface. 

Let  P  or  (x,  y,  z)  be  any  point  of  S,  a,  /?,  and  y  the  angles 
which  the  exterior  normal  drawn  at  P  to  S  makes  with  the 
coordinate  axes,  and  P'  a  point  on  this  normal  at  a  distance 
AH  from  P.  The  coordinates  of  P'  are 

#  +  A?i*cosa,    ?/  +  A>?  -cos/2,    z  -f-  An*  cosy, 

and  if  W=f(x,  y,  z)  be  any  continuous  function  of  the  space 
coordinates, 


WP>  =f(x  4-  AH  cos  a,  y  4-  AH  cos/2,  z  +  AH  cosy) 

=/(a,  y,  z)  4-  AH  cos  a  •  DJ+  AH  cos/? 
and  -f  A 


whence 

lim  TF~/7/>  =  A,  TFP  =  cosa  A/+cos/8Z>,/+cosy  Z),/.  [143] 


If,  as  a  special  case,  TK=  aj,  we  have  .Dn;e  =  cosa;  so  that 
[142]  may  be  written 

C  CCDxU-dxtlydz=  CuDnx-ds,  [144] 

which  we  were  to  prove.* 

Green's  Theorem,  which  follows  very  easily  from  this  result, 
may  be  stated  in  the  following  form  : 

If  U  and  V  are  any  two  functions  of  the  space  coordinates 
which  together  with  their  first  derivatives  with  respect  to  these 
coordinates  are  finite,  continuous,  and  single-valued  throughout 
the  space  shut  in  by  an}*  closed  surface  /S,  then,  if  n  refers  to 
an  exterior  normal, 

*  This  theorem  has  been  virtually  proved  already  in  Sections  29  and  36. 


94  SURFACE   DISTRIBUTIONS. 


=  £u-  Dn  V-  ds  -  C  C  Cu-  V2  V-  dxdydz  [145] 

=  Cv-DnU-ds-  C  C  Cv-V'2U.  dxdydz,          [146] 

where  the  triple  integrals  include  all  the  space  within  S  and  the 
single  integrals  include  the  whole  surface. 

Since          DXU>  DXV=  DX(U-  DXV)  -  U-  DX2V, 
we  have  C  C  (*DX  U-  Dx  V-  dx  dy  dz 

=  C  C  CDX(U-  DxV)dxdydz-  C  C  Cu-  DX2V>  dxdydz; 
but,  from  [144], 

C  C  CDX(  U-DxV)dxdydz  =  Cu-DxV-  Dnx  .  ds, 

whence  f  C  C(L>XU>  DXV)  dxdydz 

=  CU'DxV'Dnx-ds-  CC  Cu.Dx2V-dxdydz.     [147] 

If  we  form  the  two  corresponding  equations  for  the  deriva 
tives  with  regard  to  y  and  z,  and  add  the  three  together,  we  shall 
obtain  an  expression  which,  by  the  use  of  [143],  reduces  im 
mediately  to  [145].  Considerations  of  symmetry  give  [146]. 

If  we  subtract  [146]  from  [145],  we  get 

IT  A  U-  V2F-  V-  V2  U)dxdydz 

-DnV-  V-DnU)ds.  [148] 


In  applying  Green's  Theorem  to  such  spaces  as  those  marked 
T(}  in  the  adjoining  diagrams,  it  is  to  be  noticed  that  the  walls 
of  the  cavities,  marked  /S'  and  $",  as  well  as  the  surfaces, 


GREEN'S  THEOREM. 


95 


marked  S,  form  parts  of  the  boundaries  of  the  spaces,  and  that 
the  surface  integrals,  which  the  theorem  declares  must  be  taken 


FIG.  35. 


over  the  complete  boundaries  of  the  spaces,  are  to  be  ex 
tended  over  S'  and  S"  as  well  as  over  S.  We  must  remember, 
however,  that  an  exterior  normal  to  T0  at  S'  points  into  the 
cavity  C'. 

If  U  and  V  both  satisfy  Laplace's  Equation,  the  second 
member  of  [148]  is  equal  to  zero. 

If  within  the  closed  surface  S  the  functions  X,  U,  and  V 
are  continuous,  and  if  the  first  derivatives  of  U  and  V  are 
continuous  (the  first  derivatives  of  A  and  the  second  deriva 
tives  of  U  and  V  being  finite), 


(A  •  Dy  V)  +  D=  (A  -DZV)-]  dxdydz        [149] 


+  DV(X-  Dy  U}  +  Dz  (A  -  Dg  U)']  dxdydz. 

Special  Cases  under  Green's  Theorem.    Applications. 

I.  If  in  [145]  we  put  U=  1,  we  learn  that  if  Fis  any  function 
which  within  and  on  the  closed  surface  S  is  finite  and  contin- 


96  SURFACE    DISTRIBUTIONS. 

nous,  together  with  its  derivatives  of  the  first  order,  the  surface 
integral  of  Dn  V  taken  over  S  is  equal  to  the  volume  integral 
of  V2  V  taken  through  the  space  shut  in  by  S.  If  V  happens 
to  satisfy  Laplace's  Equation  within  S,  the  surface  integral 
is  equal  to  zero.  This  result  should  be  compared  with  Gauss's 
Theorem,  treated  in  Section  31. 

II.  If  in  [145]  we  make  £7  equal  to  F,  the  potential  function 
due  to  any  distribution  of  matter,  and  assume  that,  in  the 
general  case,  some  of  this  matter  is  spread  superficially  on  a 
surface  S  (or  on  a  number  of  such  surfaces),  we  may  shut  in 
S  by  two  other  surfaces,  Si  and  S2t  parallel  and  very  close  to 
it.  We  may  then  apply  Green's  Theorem  to  so  much  of  the 
space  within  a  spherical  surface,  with  centre  at  some  con 
venient  fixed  point  and  radius  r  large  enough  to  include 
the  whole  distribution,  as  does  not  lie  between  Si  and  S2. 
This  gives 


C  C  Cv 


\*Vdxdydz, 


where  the  first  surface  integral  is  to  be  extended  over  the 
spherical  surface,  the  second  over  S^  and  the  third  over  S2,  it 
being  understood  that  HI  represents  a  normal  to  Si  taken  in 
the  direction  away  from  S,  and  n2  a  normal  to  S2  taken  in  the 
direction  away  from  S.  Since  V  is  continuous  at  S,  while  its 
normal  derivatives  are  discontinuous  in  the  manner  indicated 
by  the  equation  D  V  -f-  D^  V  =  —  4  TTO-,  the  limit  of  the  sum 
of  the  two  surface  integrals  taken  over  Si  and  Sz  as  these 

surfaces  approach  S  is  4?r  J  Va-ds.      The  value  of  the  first 

surface  integral  is  equal  to  4  -n-r2  times  the  average  value  of 
V  •  DrV  on  the  surface  ;  and,  if  this  be  written  in  the  form  4?r 


GREEN'S  THEOKKM.  97 

[average  value  of  F(/-2Z>rF)],  it  is  evident  that  the  integral 
approaches  zero  as  the  radius  r  is  made  infinite,  so  that  the 
field  of  the  triple  integrals  may  embrace  all  space.  Since 
V-r=  —  47iy>,  the  whole  second  member  of  the  equation 

represents  4?r  lim        FA»&  extended  over  all  the  distribution, 


and  this  is  8  TT  times  the  intrinsic  energy  of  the  distribution. 
The  first  member  of  the  equation  represents  the  volume  integral 
of  the  square  of  the  resultant  force  extended  over  all  space. 
We  may  write  this  result  in  the  form 


III.  If  in  [145]  we  make  U  =  V=  n,  any  function  which 
within  the  closed  surface  S  satisfies  the  equation  Y'-V  =  0,  we 
shall  have 

D,tu  •  dS.  [151] 

IV.  If  in  [148]  V  is  the  potential  function  due  to  two  dis 
tributions  of  active  matter,  Mv  inside  the  closed  surface  S  and 

J/2  outside  it,  and  if  U=->  where  r  is  the  distance  of  the  point 


(x,  y,  z)  from  a  fixed  point  0,  we  must  consider  separately  the 
two  cases  where  0  is  respectively  without  S  and  within  S. 

A.    If  0  is  without  S,  V2  (  -  J  =  0  for  points  within  the  sur 
face.     Also,  V2  F  =  —  4  ?rp,  so  that 


98  SURFACE    DISTRIBUTIONS. 


The  triple  integral  is  evidently  equal  to  the  value  at  the  point 
0  of  the  potential  function  due  to  Mv  alone.  If  we  call  this  Fi, 
and  notice  (see  [143])  that  Dnr  at  any  point  of  S  is  the  cosine 
of  the  angle  8  between  r  and  the  exterior  normal  to  $,  we  have 


ds  +  dS  =  -  4  ,F,  [152] 

If  $  is  a  surface  equipotential  with  respect  to  the  joint  action 
of  Ml  and  Jf2>  and  if  we  denote  by  Fs.  the  constant  value  of 
V  on  S,  we  have 


and  it  is  easy  to  show,  by  the  reasoning  used  in  Section  31, 

/~cos  8 

O  >&         that  I  — —  dS  =  0,  whence 
-  .    <2  J     r* 

HlM^?  j 


B.    If  0  is  a  point  inside  $,  whether  or  not  it  is  within  MI, 
and  if  S  is  equipotential  with  respect  to  the  action  of  Ml  and 


FIG.  37. 

Jf2,  we  will  surround  0  by  a  small  spherical  surface  £'  of 
radius  r'  and  apply  [148]  to  the  space  lying  inside  S  and 
without  the  spherical  surface.  In  doing  so,  it  is  to  be  noticed 
that  S'  forms  part  of  the  boundary  of  the  region  we  are  deal 
ing  with,  and  that  an  exterior  normal  to  the  region  at  S'  will 
be  an  interior  normal  of  the  sphere. 


GREEN'S  THEOREM.  99 

Since  for  all  points  of  the  region  we  are  considering  v4  -  )=0, 


we  have 


or,  since  dS'  =  r'2dto'}  where  dot'  is  the  area  which  the  elemen 
tary  cone  the  base  of  which  is  dS'  and  the  vertex  O  intercepts 
on  the  sphere  of  unit  radius  drawn  about  O, 


cls  +  rj?»*  ds  _  ,,  J  A.  r. 


It  is  easily  proved,  by  the  reasoning  of  Section  31,  that 


f 
J 


:    4 


and  it  is  clear  that  if  r'  be  made  smaller  and  smaller,  the  third 
integral  of  [155]  approaches  the  limit  zero.  If  F'  is  the 
average  value  of  F  on  the  surface  •  S  ', 


C  Frfw'  =  F1  frfoi'  =  F'  4  TT  : 


and  as  r'  is  made  smaller  and  smaller  this  approaches  the 
value  4  ?r  F0,  where  F0  is  the  value  of  F  at  0.  The  value, 
when  r'  is  zero,  of  the  triple  integral  is  evidently  F1?  and  we 
have 


If  F2  is  the  value  at  0  of  the  potential  function  due  to  M2 
alone,  V0  =  l\  +  Fa,  so  that 


100  SURFACE    DISTRIBUTIONS. 


If  S  is  not  equipotential  with  respect  to  the  action  of  MI 
and  M2,  we  have 

";r*s~/r-Z)"(r)*s-     [154'] 

V.    If  in  [148]  we  make  U  =-•>  where  r  is  the  distance  of 

the  point  (aj,  y,  z)  from  a  fixed  point  0,  and  if  F=  v,  any  func 
tion  harmonic  everywhere  within  the  closed  surface  S,  we 
shall  have 


£-  dS,  [155A] 

I* 


if  0  is  within  £,  and 

/  V'  ^  =/'•'•  A  (~Vw,  [155,,] 


if  0  is  outside  /S. 

VI.  The  closed  surface  S  encloses  a  region  TI  and  excludes 
the  rest  of  space,  Tz.  A  function  V  is  continuous  and  has 
finite  first  and  second  derivatives  everywhere  in  the  field  of 
Green's  Theorem.  The  first  derivatives  are  everywhere  con 
tinuous  except  at  certain  surfaces,  Si  in  Tl  and  Sz'  in  TZ1 
where  the  tangential  derivatives  are  continuous,  and  the  nor 
mal  derivatives  discontinuous  in  the  manner  indicated  by  the 
equation 


At  infinity  V  vanishes  like  the  Newtonian  Potential  Func 
tion  due  to  a  finite  distribution  of  matter.  If  U  is  the  recip 
rocal  of  the  distance  from  a  fixed  point  0,  and  if  we  apply 
Green's  Theorem  to  U  and  F,  using  successively  as  fields,  5\ 
when  0  is  in  T2,  7\  when  0  is  in  Tly  Tz  when  0  is  in  Tt,  and 


GREEN'S  THEOREM.  101 

T2  when  O  is  in  jT2,  and  representing  by  n  a  normal  to  S 
pointing  into  T2  in  all  cases,  we  learn  'that  the  expression 


is  equal  respectively  to 


If  there  is  no  surface  £'  at  which  the  normal  derivative 
of  F  is  discontinuous,  and  if  F  satisfies  Laplace's  Equation 
everywhere  within  S,  the  expression 


is  equal  to  zero  or  to  the  value  of  V  at  0  according  as  0  is 
without  or  within  S. 

If,  now,  S  is  a  spherical  surface  of  radius  a,  and  if  Ox  is 
distant  Zj  from  the  centre  C,  the  distance  from  (7  of  02,  the 
inverse  point  of  Ol  with  respect  to  S  may  be  denoted  by  /2, 
where  ^  =  a2.  If  r^  and  r2  represent  the  distances  of  any 
point  P  from  Ol  and  02  respectively,  then,  if  P  lies  on  S, 


102  SURFACE    DISTRIBUTIONS. 


42  =  r22  +  a2  —  2  r2a  •  cos  (r2,  n) 

cos  (/-!,  n)       a    cos  (r2,  ?i)  _  a'2  — 
y*!2  li          r22  a^' 

In  this  case, 


and 


so  that  it  is  easy  to  eliminate  Dn  V  by  multiplying  the  second 
equation  by  a/h  and  subtracting  the  members  from  those  of 
the  first  equation.  The  result  is 


0 


1     rr  _  F(a2  -  Zt2)  ^^ 
~  4  Tra  J  J  [a2  +  /i2  -  2  ^  cos  (a,  /i)]8/2 


.This  integral  determines  F  at  every  point  within  S  when 
its  value  is  given  at  every  point  on  S.     If  Ox  is  at  the  centre 


of 


S,  I,.  =  0,  and  ^  =  a,  so  that  V0  =  -  -  5  f  |Fc?^,  or  the 

4  7T(l   J   J 

average  value  on  a  spherical  surface  S,  of  a  function  F,  har 
monic  within  and  on  S  is  the  value  of  V  at  the  centre  of  & 
It  follows  from  this  that  a  function  which  is  harmonic  about 
a  point  0  cannot  have  at  0  either  a  maximum  or  a  minimum 
value. 


GREEN'S  THEOREM.  103 

If  a  function  V  is  constant  on  any  analytic  surface  S,  is 
harmonic  without  S,  and  if  it  vanishes  at  infinity  like  a  New 
tonian  Potential  Function, 


and  V  is  the  potential  function  in  outer  space  due  to  a  super 
ficial  distribution  on  S  of  surface  density  —  DnV/±ir. 

VII.  A  function  V  has  the  value  zero  everywhere  on  the 
closed  surface  Si,  and  the  constant  value  C  on  the  closed  sur 
face  S2,  shut  in  by  S^  In  the  space  T,  between  Si  and  S9,  V 
is  harmonic.  If  we  apply  Green's  Theorem,  in  T,  to  V  and 
to  the  reciprocal  of  the  distance  from  any  point  0  in  T,  we 
learn  that 


where  both  normals  point  out  of  T. 

V  is,  therefore,  the  potential  function  due  to  surface  distri 
butions  on  Si  and  Sz  numerically  equal  to  DHV/±v  at  every 
point. 

VIII.  If  the  closed  surface  S  shuts  in  a  region  T,  and  if 
the  functions  V  and  V\  which  are  equal  at  every  point  of  S, 
are  finite  and  continuous  with  their  derivatives  of  the  first 
order  at  every  point  of  T,  and  if  within  T,  V  does  and  V 
does  not  satisfy  Laplace's  Equation,  then  the  integral 


a  +  (*>*  *T  +  (A  *T]  dxdydz, 
extended  throughout  T  is  less  than  the  corresponding  integral 

Qy  '^ttWy  +  (WY'  +  (W 


If  we  write  V  =  V  +  u,  u  vanishes  at  every  point  of  S,  but 
is  in  general  different  from  zero. 


104  SURFACE    DISTRIBUTIONS. 


=  Q  v+  Qu  +  2  [D*u  •  D*  V+E>J<<  •  I>v  V+Dsu  •  D 

2  C  Cn-DnVdS-2  C  C  C  u-^Vdxdydz 


Now,  since  the  integrands  of  Qu  and  $rare  made  up  of  squares, 
and  since  neither  u  nor  V  are  constants,  both  Qu  and  Qv  are 
positive,  so  that  Qv-  >  Qv. 

IX.  There  cannot  be  two  different  functions,  Wl  and  W^ 
which  have  equal  values  at  every  point  of  Si  and  Sz  (two 
closed  surfaces  the  first  of  which  shuts  in  the  second),  and 
between   these    surfaces    are    everywhere   harmonic.      If  we 
suppose,  for  the  sake  of  argument,  that  two  such  functions 
exist  and  call  their  difference  u,  it  is  clear  that  u  is  harmonic 
between  the  surfaces  and  that  it  vanishes  at  every  point  of 
both  Si  and  $2.     If,  therefore,  in  [145]  we  make  U  —  V  =  u, 
we  learn  that 

2]  dxd>jdz  =  0, 

where  the  integral  extends  over  all  the  space  between  Si  and 
$2.  Since  the  integrand  cannot  be  negative,  it  must  be  zero-  at 
every  point,  so  that  Dxu  =  Dyu  =  Dzu  =  0  and  u  is  constant. 
But  u  =  0  on  Su  therefore  it  is  identically  equal  to  zero  and 
Wi  =  W» 

It  is  easy  to  show  that  two  functions  which  have  equal 
normal  derivatives  at  every  point  of  Si  and  $2,  and  are  har 
monic  everywhere  between  the  surfaces,  can  differ  only  by  a 
constant. 

X.  We  may  now  give  an  old  proof  of  a  theorem,  originally 
discovered  by  Green  from  physical  considerations,  which  is 
usually  called  Dirichlet's   Principle  by  Continental  writers, 


GREEN'S  THEOREM.  105 

but  in  English  books  is  generally  attributed  to  Sir  W.  Thom 
son.*  This  theorem  asserts  that  there  always  exists  one,  but 
no  other  than  this  one,  function,  v,  of  x}  y,  z,  which  (1)  is 
continuous,  and  single-valued,  together  with  its  first  space 
derivatives,  throughout  a  given  closed  region  T ;  (2)  at  every 
point  of  the  region  satisfies  the  equation  V2  v  =  0  ;  and  (3)  at 
every  point  on  the  boundary  of  the  region  has  any  arbitrarily 
assigned  value,  provided  that  this  can  be  regarded  as  the  value 
at  that  point  of  a  single-valued  function,  continuous  all  over 
this  boundary. 

There  is  evidently  an  infinite  number  of  functions  which 
satisfy  the  first  and  third  conditions.  If,  for  instance,  the  equa 
tion  of  the  bounding  surface  S  of  the  region  is  F(x,  y,  z)  =  0, 
and  if  the  value  of  v  at  the  point  (x,  y,  2)  upon  this  surface  is 
to  be  f(Xj  y,  z),  any  function  of  the  form 

3>(a-,  y,  z)-F(x,  y,  z)  +f(x,  y,  z) 

would  satisfy  the  third  condition,  whatever  continuous  function 
<£  might  be. 

If  we  assign  to  the  function  to  be  found  a  constant  value  C 
all  over  S,  v  =  C  will  satisfy  all  three  of  the  conditions  given 
above. 

If  the  sought  function  is  to  have  different  values  at  different 
points  of  S,  and  if  for  u  in  the  integral 

Q  = 

which  is  to  be  extended  over  the  whole  of  the  region,  we  sub 
stitute  any  one  of  all  the  functions  which  satisfy  conditions 
(1)  and  (3),  the  resulting  value  of  Q  will  be  positive.  Some 
one  at  least  of  these  functions  (*;)  must,  however,  yield  a 
value  of  Q  which,  though  positive,  is  so  small  that  no  other 
one  can  make  Q  smaller. f  Let  h  be  an  arbitrary  constant  to 

*  W.  Thomson,  Lioumlle's  Journal,  1847.  Dirichlet's  Vorlesungen. 
Bacharach,  Abriss  der  Geschichte  der  Potent ialtheorie. 

t  A  principle  which  will  doubtless  lead  to  a  justification  of  this  by  no 
means  self-evident  assumption  was  pointed  out  by  Hilbert  in  a  remark 
able  paper  read  before  the  Deutsche  Matheinatiker-Vereinigung  in  1899. 


106  SURFACE    DISTRIBUTIONS. 

which,  some  value  has  been  assigned,  and  let  w  be  any  func 
tion  which  satisfies  condition  (1)  and  is  equal  to  zero  at  all 
parts  of  S,  then  U  =  v  +  hw  will  satisfy  conditions  (1)  and 
(3),  and,  conversely,  there  is  no  function  which  satisfies  these 
two  conditions  which  cannot  be  written  in  the  form  U=  v  +  hw, 
where  h  is  an  arbitrary  constant,  and  w  some  function  which 
is  zero  at  S  and  which  satisfies  condition  (1). 

Call  the  minimum  value  of  Q  due  to  v,  Qv)  and  the  value  of 
Q  due  to  U,  QUt  then 


which,  since  w  is  zero  at  the  boundary  of  the  region,  may  be 
written,  by  the  help  of  Green's  Theorem, 


u-Qv  =  -2hCC  CwV>2vdxdyd 


Now,  since  Qv  is  the  minimum  value  of  Q,  no  one  of  the 
infinite  number  of  values  of  Qv  —  Qv  formed  by  changing  h 
and  w  under  the  conditions  just  named  can  be  negative  ;  but 
if  V2u  were  not  everywhere  equal  to  zero  within  T,  it  would 
be  easy  to  choose  w  so  that  the  coefficient  of  2  A  in  the  expres 
sion  for  Qu  —  Qv  should  not  be  zero,  and  then  to  choose  h  so 
that  Qu  —  Qv  should  be  negative.  Hence  V2t>  is  equal  to  zero 
throughout  T,  and  there  always  exists  at  least  one  function 
which  satisfies  the  three  conditions  stated  above.  Compare 
VIII. 

There  is  only  one  such  function ;  for  if  beside  v  there  were 
another  u  =  v  4-  hw,  we  should  have,  since  the  coefficient  of  h 
is  zero  when  V2w  =  0, 

and  that  Qu  may  be  as  small  as  Qv,  h&  must  be  zero,  whence 
either  h  =  0  or  Q  =  0,  and  if  O  =  0,  iv  is  zero.     Therefore, 


GREEN'S  THEOREM.  107 

u  —  r,  and  there  is  only  one  function  which  in  any  given  case 
satisfies  all  the  three  conditions  given  above. 

XI.  The  potential  function  F,  due  to  a  volume  distribution 
of  finite  density  p  in  the  region  T  and  a  superficial  distribu 
tion  of  finite  surface  density  o-  on  the  surface  S,  is  everywhere 
continuous,  and  it  so  vanishes  at  infinity  that,  if  r  is  the  dis 
tance  from  any  finite  point,  each  of  the  quantities 

rV,    -r2DrV, 

as  r  becomes  infinite,  approaches  the  limit  Mt  where  M  is  the 
amount  of  matter  (algebraically  considered)  in  the  whole  dis 
tribution.  The  first  derivatives  of  V  are  everywhere  finite, 
and  they  are  continuous  except  on  S,  at  every  point  of  which 
tangential  derivatives  are  continuous,  while  the  normal  deriva 
tive  is  discontinuous  in  the  manner  indicated  by  the  equation 


where  »i  and  n2  are  the  normals  to  the  surface  drawn  away 
from  it  on  each  side.  The  second  derivatives  of  V  are  every 
where  finite,  and  they  are  continuous  except  at  surfaces  where 
p  is  discontinuous.  At  any  point  on  such  a  surface  the  tan 
gential  second  derivatives  are  continuous,  but  the  normal  sec 
ond  derivative  is  discontinuous  by  an  amount  equal  to  4  TT 
times  the  discontinuity  in  p  reckoned  in  the  direction  opposite 
to  that  in  which  the  derivative  is  taken.  Everywhere,  except 
at  surfaces  of  discontinuity  in  p,  V  satisfies  Poisson's  Equa 
tion,  V2  V  =  —  4  -n-p,  and  without  T,  where  there  is  no  matter, 
this  degenerates  into  Laplace's  Equation. 

For  a  given  value  of  p  in  the  given  region  T,  and  a  given 
value  of  a-  on  the  given  surface  S,  only  one  function  has  all 
these  properties.  Assuming  that  there  are  two  such  functions, 
V  and  F',  let  their  difference  be  the  function  u.  At  every 
point  of  S, 

A,  v  +  A,,  v  --=  A,  rf  +  A,  v  =  -  4  TTO-, 

so  that  2)nu  +  Dnu  =  0, 


108  SURFACE    DISTRIBUTIONS. 

and  even  the  normal  derivatives  of  u  are  continuous  at  every 
point  of  S.  At  surfaces  of  discontinuity  in  p,  the  derivatives 
of  u  are  all  continuous  and  u  satisfies  everywhere  Laplace's 
Equation.  The  limits,  as  r  becomes  infinite,  of  ru  and  r*Dru 
are  zero.  Since  u  with  its  first  and  second  derivatives  is 
everywhere  continuous,  we  may  imagine  a  spherical  surface 
of  large  radius  r,  drawn  about  any  finite  point  0,  as  centre, 
so  as  to  enclose  all  the  attracting  mass  and  apply  Green's 
Theorem  in  the  form  of  [151]  to  u  inside  this  surface.  The 
numerical  value  of  the  surface  integral 


uD.udS 


taken  over  the  spherical  surface  is  no  greater  than  the  area 
of  the  surface  (4?rr2)  multiplied  by  the  largest  value  which 
u  •  Dru  has  on  the  surface,  or 

4?r  [greatest  value  of  (i(,r1Dru)~]. 

If,  now,  the  radius  of  the  surface  be  indefinitely  increased, 
this  expression  approaches  the  limit  zero  so  that  the  integral 


taken  over  all  space  has  the  value  zero.  Since  the  integrand 
is  made  up  of  squares  which  can  never  be  negative,  we  must 
have  at  every  point  of  space 

Dxu  =  Dyu  =  DM  =  0. 

Therefore,  u  is  constant  in  all  space  ;  and  since  it  is  zero  at 
infinity,  it  must  be  everywhere  zero,  so  that  V  and  V  are 
identical.  It  is  to  be  understood  that  T  may  be  made  up  of 
several  distinct  regions,  and  that  S  may  consist  of  several 
distinct  surfaces. 


GREEN'S  THEOREM.  109 

49.  The  Surface  Distributions  Equivalent  to  Certain  Volume 
Distributions.  Keeping  the  notation  of  IV.  in  the  last  article, 
let  S  be  a  closed  surface  equipotential  with  respect  either  to 
the  joint  action  of  two  distributions  of  matter,  Ml  inside  S  and 
J/2  outside  it,  or  (when  J/2  equals  zero)  to  the  action  of  a 
single  distribution  within  the  surface  ;  and  let  Fj,  F"2,  and  V 
be  the  values  of  the  potential  functions  due  respectively  to  J/i 
alone,  to  J/2  alone,  and  to  J/t  and  M2  existing  together.  If  a 

quantity  of  matter  were  condensed  on  S  so  as  to  give  at  every 

2)  y 

point  a  surface  density  equal  to — ,  the  whole  quantity  of 

4?r 

matter  on  the  surface  would  be 


and  this,  by  §  31,  is  equal  in  amount  to  J/i.  Let  us  study  the 
effect  of  removing  3/i  from  the  inside  of  S  and  spreading  it  in 
a  superficial  distribution  J/i'  over  S,  so  that  the  surface  density 

—  D  V 

at  every  point  shall  be —     In  what  follows,  it  is  assumed 

4?r 

that  we  have  two  distributions  of  matter,  one  inside  the  closed 
surface  and  the  other  outside.  It  is  to  be  carefully  noted,  how 
ever,  that  by  putting  J/2  equal  to  zero  in  our  equations,  all  our 
results  are  applicable  to  the  case  where  we  have  an  equipotential 
surface  surrounding  all  the  matter,  which  may  be  all  of  one  kind 
or  not. 

The  value,  at  any  point  0,  of  the  potential  function  due  to 
the  joint  effect  of  J/2  and  the  surface  distribution  J/i',  would  be 

F 


v  _  v        1    CD, 

>  o  —  y\  — ; —  I  — 

4  -. "      / 


dS. 


If  0  is  an  outside  point,  we  have,  by  [153], 


so  that  the  effect  at  any  point  outside  an  equipotential  surface 
of  a  quantity  J/j  of  matter  anyhow  distributed  inside  the  sur 
face  is  the  same  as  that  of  an  equal  quantity  of  matter  dis 
tributed  over  the  surface  in  such  a  way  that  the  superficial 


110  SURFACE   DISTRIBUTIONS. 

_  2)  y 

density  at  every  point  is  —     w    ,  where  V  is  the  value  of  the 

4  IT 

potential  function  due  to  the  joint  action  of  MI  and  any  matter 
(M2)  that  may  be  outside  the  surface. 
If  0  is  an  inside  point,  we  have, 


which  shows  that  the  joint  effect  of  M2  and  J//  is  to  give  to  all 
points  within  and  upon  the  surface  the  same  constant  value  of 
the  potential  function  which  points  upon  the  surface  had  before 
J/i  was  displaced  by  M{'.  If,  therefore,  MJ  and  M2  exist  without 
J/i,  there  is  no  force  at  any  point  of  the  cavity  shut  in  by  $; 
or,  in  other  words,  the  force  due  to  MJ  alone  is  at  all  points 
inside  S  equal  and  opposite  to  that  due  to  M2. 

If  J/i  and  M2  exist  without  MJ,  the  cavity  enclosed  by  S  is,  in 
general,  a  field  of  force.  MI  acts  as  a  screen  to  shield  the  space 
within  S  from  the  action  of  M2. 

The  surface  of  M^  is  equipotential  with  respect  to  all  the 
active  matter,  so  that  there  is  no  tendency  of  the  matter  com 
posing  the  surface  distribution  to  arrange  itself  in  any  different 
manner  upon  S. 

Since  M^  exerts  the  same  force  on  every  particle  outside 
S  that  M-i  did,  and  since  action  and  reaction  are  equal  and 
opposite,  every  particle  of  M2  exerts  on  MI  forces  the-  result 
ant  of  which  is  equal  to  the  resultant  of  the  forces  with 
which  the  same  particle  urged  M^  The  resultant  effect, 
therefore,  of  the  action,  of  M2  on  If/  is  the  same  as  the 
resultant  effect  of  its  action  on  Mlt  Now  the  whole  system 
of  forces  applied  to  the  surface  distribution  by  Mz  and  by 
the  repulsions  for  one  another  of  its  own  parts  is  equivalent 
to  a  tension  from  without  of  2  no-'2  dynes  per  square  centi 
meter  applied  all  over  S,  and  since  the  internal  forces  form 
a  system  in  equilibrium,  the  resultant  effect  of  Mz  on  Ml 
is  equal  to  the  resultant  effect  of  the  tension  just  mentioned 
on  MJ. 

If  two  closed  surfaces,  ^  and  S2,  which  mutually  exclude 
each  other,  shut  in,  respectively,  the  two  portions,  M1}  M2)  of  a 


GREEN'S  THEOREM.  Ill 

distribution  3/,  and  are  level  surfaces  of  M's  potential  function, 
it  is  easy  to  see  that  a  superficial  distribution  on  Si  of  density 
a-  =  —  Dn  F/4  TT  would  act  on  a  particle  without  Si  just  as  MI 
does,  and  that  a  similar  distribution  on  S.>  would  act  on  parti 
cles  outside  of  S2  as  Mz  does.  The  action  of  J/x  on  Mz  is  the 
same  as  the  resultant  effect  of  the  tension  2  -n-a-'2  or  (Dn  F)2/8?r 
considered  as  acting  all  over  Sz.  The  surface  integral  of 
—  DnV/4:7r  extended  over  any  closed  surface  has  been  called 
by  Maxwell  the  "  electric  displacement "  through  the  surface. 

50.  Vectors.  Stokes's  Theorem.  The  Derivatives  of  Scalar 
Point  Functions.  It  is  frequently  convenient  to  define  a 
vector  by  giving  the  values  (tensors)  of  its  components  paral 
lel  to  the  coordinate  axes  ;  and  if  for  our  present  purposes  we 
call  these  "  the  components  of  the  vector,"  no  confusion  will 
arise.  The  expression  (Qr,  Qy,  Q^  denotes  a  vector,  Q,  the 
components  of  which  parallel  to  the  axes  of  x,  ?/,  and  z  are 
respectively  equal  to  Qx,  Qy,  and  Qz.  The  direction  cosines  of 
the  vector  are  the  ratios  of  Qx,  Qy,  Qz  to  V ' Qx2  +  Qy2  +  Q*. 
The  letter  which  represents  a  vector  is  often  used  in  scalar 
equations  to  denote  merely  the  tensor.  Sometimes,  however, 
the  heavy  face  letter  (Q)  is  used  to  denote  the  vector,  while 
its  tensor  is  represented  by  the  same  letter  in  ordinary  type. 
Any  three  scalar  point  functions  can  be  considered  the  com 
ponents  of  a  vector  point  function.  Scalar  and  vector  point 
functions  are  sometimes  called  "  distributed "  scalars  and 
vectors.  Where  there  is  no  danger  of  any  misunderstanding 
a  vector  point  function  may  be  called  simply  a  vector. 

The  scalar  function  DxQr-\-  DvQy-\-DzQs  is  called  the  diver 
gence  of  Q,  and  if  this  quantity  vanishes  identically,  Q  is  said 
to  be  a  solenoidal  vector.  The  force  due  to  any  finite  distri 
bution  of  matter  attracting  or  repelling  according  to  the  "  Law 
of  Nature  ''  is  solenoidal  in  empty  space.  The  negative  of  the 
divergence  of  a  vector  is  called  its  convergence. 

The  vector,  the  components  of  which  taken  parallel  to  the 
coordinate  axes  are  the  three  scalar  point  functions, 

Aft  -  Aft,,  Aft  -  Aft,  Aft  -  Aft. 


112  SURFACE    DISTRIBUTIONS. 

is  called  the  curl  of  Q\  and  if  these  components  vanish  at  every 
point  of  a  region,  Q  is  said  to  be  lamellar  in  that  region.  If 
the  vector  R  is  the  curl  of  the  vector  Q,  Q  is  said  to  be  a 
vector  potential  function  of  R.  The  force  due  to  a  finite  dis 
tribution  of  attracting  or  repelling  matter  is  lamellar  within 
and  without  the  distribution.  The  curl  of  any  vector  is 
itself  solenoidal.  If  two  vectors  have  the  same  curl,  their 
difference  is  a  lamellar  vector. 

The  lines  of  a  vector  are  a  family  of  curves,  one  of  which 
passes  through  every  point  of  space,  and  each  of  which  has 
at  every  one  of  its  points  the  direction  of  the  vector  at  the 
point.  The  differential  equations  of  the  lines  of  the  vector  Q 
are  evidently  dx  /  QX  =  dy  /  ^  =  dz  /  Q  . 

after  the  values  of  Qx,  Qv,  and  Qz  have  been  substituted,  we 
have  two  equations  of  the  form 

dx  dy 

—  =  <f>(x,  y,  z),  -^  =  if/(x,  y,  «); 

whence  we  get,  by  differentiating, 

d2x  dx 

—  =  DA  -^  +  DA  •  *  +  Dz*, 


and,  by  eliminating  y  between  the  first  and  third  equations, 
and  x  between  the  second  and  fourth  equations,  two  equations 
of  the  second  order  between  x  and  z  and  between  y  and  z 
respectively.  The  integrals  of  these  last  equations  are  the 
equations  of  the  lines  of  the  vector.  Sometimes  the  variables 
may  be  separated  at  the  start,  and  then  the  work  is  much 
simplified.  The  lines  of  the  vector  (—  x2,  y,  z)  have  the  equa 
tions  y  =  Az,  a?log(2?y)=l,  and  those  of  the  vector  (3xz  —  yz, 
xz  +  yz,  «),  the  equations  x  =  (B  +  A  -f-  Bz)  e*z,  y  —  (A-\-  Bz)  e2z, 
where  A  and  B  are  arbitrary  constants. 

If  n  represents  the  exterior  normal  of  any  closed  surface  S, 
the  integral  taken  over  S  of  the  exterior  normal  component 
of  the  analytic  vector  Q  is 


GREEN'S  THEOREM.  113 

f#cos(n>  Q)dS 

=  CQ  [cos  (x,  Q)  cos  (x,  n)  +  cos  (#,  Q)  cos  (y,  w) 

+  cos  (z,  Q)  cos  (2,  71)]  d£ 
=  f[<?x  cos  (x,  n)  +  Qy  cos  (?/,  7i)-h  $2  cos  (2,  7i)]cZS; 

and  this  is  equal  to  the  volume  integral  of  the  divergence  of 
Q  taken  through  the  space  within  S.  The  integral  of  the 
exterior  normal  component  of  any  analytic  solenoidal  vector, 
taken  over  any  closed  surface,  is  zero. 

An  important  theorem  due  to  Sir  George  Gabriel  Stokes 
may  be  stated  as  follows  : 

The  line  integral  taken  around  a  closed  curve  s,  of  the  tan 
gential  component  of  an  analytic  vector  point  function  Q,  is 
equal  to  the  surface  integral  taken  over  any  surface  S,  bounded 
by  the  curre,  of  the  normal  component  of  the  curl  of  the  vector, 
the  direction  of  integration  around  the  curve  forming  a  right- 
handed  screw  rotation  about  the  normals,  or 

J  [ Qx  cos  (x,  s)  +  Qy  cos  (y,  s)  +  Qz  cos  (z,  s)~]  ds 


Aft  -  A^)  cos  (*,  n) 

+  (A<?*-^£S)COSO/,  71) 

-f  (DxQy  -  DyQx)  cos  (z,  n)-]  dS.  [158] 

To  prove  this,  we  may  evaluate  first  so  much  of  the  double 
integral  as  involves  Qx,  that  is, 

\DZQX  •  cos  (y,  71)  -  DyQx  •  cos  (z,  n)-]dS. 


Let  the  area  S  be  divided  into  quadrilateral  elements  by 
means  of  equally  spaced  planes  parallel  to  the  planes  of  zy 
and  xy  respectively,  and  consider  especially  one  of  these 
elements,  \S,  the  projection  of  which  on  the  xz  plane  is  Ax  •  Az, 
so  that  \S>  cos  (?/,  n)  =  Ax  •  Az  approximately. 


114 


SURFACE   DISTRIBUTIONS. 


That  corner  of  the  element  A$  which  has  the  least  x  and  g 
coordinates  shall  be  the  point  P}  and  that  side  of  the  element 
which  passes  through  P  and  is  parallel  to  the  plane  of  yz 
shall  be  represented  by  As^  Since  A^  is  perpendicular  both 
to  the  normal  to  S  at  P  and  to  the  axis  of  x,  cos  (x,  Sj)  =  0, 

and     cos  (n,  Si)  =  cos  (x,  ri)  •  cos  (x,  s^)  +  cos  (y,  ri)  -  cos  (y,  s^ 
+  cos  (z,  ri)  •  cos  (g,  Sj)  =  0, 

cos  (g,  w)  •  cos  (z, 
COS  (y,  7l) 


or 


=  -  cos  (y, 


FIG.  38. 


Moreover,  2)^  =  0  +  (DyQx)cos(y,  81)-\-(DzQx)cos(z, 


and      dS  = 


dx  dz 

-  -  -  - 
cos  (y,  n) 


dx  dz  dsl 

N 
-L  cos  (y,  n) 


. 
=  ds^dx 


cos  (z.  Si) 
-  ;  ( 
cos  (y,  n) 


GREEN'S  THEOREM.  115 

Hence,    J    J  \_DZ Qr  •  cos  (y,  w)  —  Z>v (^r  •  cos  (2,  ?i) ]  dS 

[cos(y,  ri)  •  cos  (2,  s^)DzQx  —  cos  (g,  ??)cos(^,g1)Dy^J.]^1</ar 
cos(?/,  ?;) 


-//i 

/•  /• 

=  I    I  ^>  Q 
JJ      *    r 


ftt'  COS  (*>   S0  +  DyQz'  COS 


If  we  perform  the  integration  with  respect  to  sl  and  intro 
duce  the  limits,  it  will  appear  that  this  integral  may  be  found 
by  proceeding  around  the  contour  s  in  the  direction  indicated 
in  the  theorem  and  determining  the  line  integral  of 

dx 

Qx  —  ds  =  Qx  •  cos  (xt  s)  ds, 

where  ds  is  an  element  of  s.  If  we  treat  in  a  similar  manner 
those  portions  of  the  double  integral  which  involve  Qy  and  Qz, 
the  theorem  will  be  evident. 


According  to  the  definition  used  in  the  preceding  sections, 
the  numerical  value  of  the  directional  derivative  of  any  scalar 
point  function  u,  at  any  point  P,  in  any  fixed  direction  PQ'} 
is  the  limit,  as  PQ  approaches  zero,  of  the  ratio  of  UQ  —  upto 
PQ,  where  Q  is  a  point  on  the  straight  line  PQ'  between  P 
and  Q'.  The  gradient  hu  of  the  function  u  at  P  is  the  direc 
tional  derivative  of  u  at  P  taken  in  the  direction  in  which 
u  increases  most  rapidly.  This  direction  is  normal  to  the 
surface  of  constant  u  which  passes  through  P. 

hu*  =  (Dxuy  +  (Dyu)z  +  (Dzu)*. 

The  directional  derivative  of  any  scalar  point  function 
at  any  point  in  any  given  direction  is  evidently  equal  to 
the  product  of  the  values  of  the  gradient  and  the  cosine  of 
the  angle  between  the  given  direction  and  that  in  which  the 
function  increases  most  rapidly. 

The  vector,  the  components  of  which  parallel  to  the  coordi 
nate  axes  are  numerically  equal  to  Dxu,  Dyu,  Dzu,  has  been 


116  SURFACE    DISTRIBUTIONS. 

called  the  vector  differential  parameter  of  u.  The  numerical 
value  (tensor)  of  this  vector  at  any  point  is  the  gradient  of  u 
at  the  point,  according  to  some  writers  ;  others  use  "gradient " 
to  represent  the  vector  itself.  The  lines  of  the  vector  are 
curves  which  cut  orthogonally  the  surfaces  of  constant  u,  that 
is,  the  family  of  surfaces  the  equation  of  which  is  u  =  c,  where 
c  is  a  parameter  constant  for  any  one  surface  of  the  family. 

If  f(x,  y,  z)  is  any  scalar  point  function,  any  vector  func 
tion  the  lines  of  which  cut  the  surfaces  of  constant  f  normally 
must  have  components  R  •  Dxf,  R  •  Dyf,  R  •  Dzf,  where  R  is 
some  function  of  x,  y,  and  z.  The  curl  of  this  vector  has 
components  Dzf-DyR-Dyf-DzR,  Dxf-DzR-DJ-DxR, 
Dyf-  DJR,  —  Dxf-  DyR,  and  the  cosine  of  the  angle  between 
the  vector  and  its  curl  is  zero,  so  that  these  two  vectors  are  per 
pendicular  to  each  other.  If  a  vector  has  a  curl  which  is  not 
perpendicular  to  it  at  every  point,  no  family  of  surfaces  exists 
the  members  of  which  cut  the  lines  of  the  vector  orthogonally 
at  every  point  of  space.  Every  plane  vector  point  function 
has  for  its  curl  a  vector  perpendicular  to  its  plane.  The 
vector  (3  yz,  xz,  xy)  is  not  lamellar,  but  it  is  perpendicular  to 
its  curl :  its  lines  cut  orthogonally  the  family  of  surfaces 
xzyz  =  c,  as  do  the  lines  of  the  lamellar  vector  (3  x*i/z,  xzz,  xst/), 
each  component  of  which  is  x2  times  the  corresponding 
component  of  the  first. 

If  the  ratios  of  the  corresponding  components  of  two  vector 
point  functions  are  all  equal  to  the  same  scalar  point  function, 
the  vectors  have  the  same  lines.  Two  lamellar  vectors  may 
have  the  same  lines,  thus :  the  lines  of  every  vector  of  the 
form  [_f(x)j  0,  0,]  are  parallel  to  the  axis  of  x,  and  every  such 
vector  is  lamellar,  whatever  analytic  function /may  represent. 

We  may  define  the  numerical  value  of  the  normal  deriva 
tive  at  any  point  P  of  a  scalar  point  function  u,  taken  with 
respect  to  another  scalar  point  function  v,  to  be  the  limit,  as 
PQ  approaches  zero,  of  the  ratio  of  UQ  —  u,,  to  VQ  —  vp,  where 
Q  is  a  point  so  chosen  on  the  normal  at  P  of  the  surface  of 
constant  v  which  passes  through  P  that  VQ  —  vp  is  positive. 


GREEN'S  THEOREM.  117 

If  (u,  v)  denotes  the  angle  between  the  directions  in  which  u 
and  v  increase  most  rapidly,  the  normal  derivatives  of  u  with 
respect  to  r,  and  of  v  with  respect  to  u,  may  be  written 

hH  •  cos  (u,  v)  /hv  and  hv  •  cos  (11,  v)  /  hu 

respectively.     If  hu  =  hv,  these  derivatives  are  equal. 

The  derivative  of  xyz  with  respect  to  x  +  y  +  z  has  at  the 
point  (1,  2,  3)  the  value  11/3.  The  derivative  at  the  same 
point  of  x  +  t/  +  z  with  respect  to  xyz  is  11  /49. 

51.  The  Attraction  of  Ellipsoids.  If  we  transform  the 
equation 

*».**.  *« 


to  parallel  axes,  using  a  point  A0,  which  lies  on  the  surface 
and  has  the  coordinates  (—  ar0,  —  y0t  —  z0)  as  origin,  and  then 
denote  by  0  the  angle  which  any  radius  vector  drawn  through 
A0  makes  with  the  x  axis,  the  equation  of  the  surface  in  polar 
coordinates  takes  the  form 

/cos20      sin2  0  cos2  <^       sin2  0  sin- 
v     d~  b~  c~ 


_  0  / 
"     \ 


•0  cos  0      ?/0  sin  0  cos  <f>      ZQ  sin  0  sin 

— » 1-  ' ^ 1 o 


If  AQ  were  at  that  extremity,  A,  of  the  «  axis  which  has  the 
coordinates  (—  o,  0,  0),  the  equation  would  be 


/cos2 
V    a2 


2  0      sin2  6  cos2  <ft      sin'2  0  sin2 
~~  ~~ 


we  will  denote  the  coefficient  of  R  in  this  equation  by  12(0,  0). 
Let  us  compare  the  x  components  of  the  attraction  at  A0  and 
at  A,  due  to  a  homogeneous  ellipsoid  of  density  p  bounded  by 
this  surface.  If,  with  each  of  these  points  as  origin,  a  set  of 
(conical)  surfaces  of  constant  9  with  the  constant  difference 
A0,  and  a  set  of  (plane)  surfaces  of  constant  <£  with  the  con 
stant  difference  A<£,  be  imagined  drawn,  the  ellipsoid  will  be 


118  SURFACE    DISTRIBUTIONS. 

divided  into  elementary  "  cones  "  in  two  ways.  The  vertices 
of  all  the  cones  of  one  system  will  be  A,  and  the  vertices  of  all 
the  cones  of  the  other  system  will  be  A0.  To  every  cone  of  the 
first  system  corresponds  a  cone  with  parallel  axis  belonging 
to  the  second  system,  but  whereas  every  cone  of  the  first 
system  yields  a  positive  contribution  to  the  x  force  compo 
nent  at  A,  some  of  the  corresponding  cones  of  the  second 
system  yield  negative  components  to  the  corresponding  force 
component  at  A0. 

We  shall  find  it  convenient  to  write  in  parentheses  after  R 
and  r1  the  value  of  0  and  <£  to  which  they  belong,  and  to  note 
that  r\n_e>  n  +  ^  =  -  r\e^}. 

If  the  values  of  0  and  <f>  which  correspond  to  a  given  cone 
of  the  first  system  are  00  and  <£0j  the  values  of  0  and  <£  which 
belong  to  the  corresponding  cone  of  the  second  system  may 
be  either  00  and  <£0,  or  ?r  —  00  an(i  v  +  <£o-  The  contribution  of 
any  cone  of  the  first  system  to  the  x  component  of  the  force  at 
A  is 


•f 


cos  0 
r2  sin  OdrdOd<j>  •  — —  =  PR(9t  ^  sin 


and  the  contribution  of  the  corresponding  cone  of  the  second 
system  to  the  x  component  of  the  force  at  AQ  is  either 

or   pr\v-6t  ^  +  7r)sin  0  cos  OdOdfa 


as  the  case  may  be. 

If,  now,  we  group  together  two  cones  of  the  first  system 
corresponding  to  (00,  <£0)  and  (00,  TT  +  <£0)  respectively,  we 
may  write  the  positive  contribution  coming  from  this  pair  in 

the  form 

4  cos2  00 


The  values  of  0  and  <£  for  the  corresponding  cones  of  the 
second  system  are  one  of  the  pairs 

(0o>  $0  5    ^o?  ""  +  <£(>)>    (^o?  ^o  5    v  ~~  &M  ^o)? 
(TT  —  00,  TT  +  </>0  ;   00,  TT  +  <^>0)?   or    ("•  ~  ^o?  w  +  ^o  j  ^  ~~  ^o?  <£o)- 


GREEN'S  THEOREM.  119 

The  two  values  of  6  and  <£  of  either  of  these  pairs  give  equal 
and  opposite  values  to 

(  ?/0  sin  0  cos  <f>      ZQ  sin  0  sin  <j>\ 
~~~  ~*  C( 


so  that  the  positive  contributions  of  this  pair  of  cones  of  the 
second  system  is 


This  contribution  to  the  x  component  of  the  force  at  A0  is  to 
the  contribution  of  the  corresponding  cones  of  the  first  sys 
tem  to  the  corresponding  force  component  at  A  as  x0  to  a. 
Therefore,  the  x  component  at  A0  of  the  attraction  due  to  the 
whole  ellipsoid  is  to  the  corresponding  component  at  A  as  x0 
to  a. 

If,  then,  we  know  the  values  (Xlt  Y1}  Z^  of  the  attraction 
due  to  a  homogeneous  ellipsoid  bounded  by  the  surface 


at  points  on  the  surface  at  the  negative  extremities  of  the 
semiaxes  a,  b,  c,  we  may  find  the  numerical  values  of  the  com 
ponents  parallel  to  the  coordinate  axes  of  the  attraction  at  any 
point  (—  xw  —  y0,  —  z0)  on  the  surface  from  the  equations 


The  attraction  X±  at  A  can  be  easily  found*  by  adding 
together  the  contributions  coming  from  all  the  elementary 
cones  with  vertices  at  A  into  which  the  ellipsoid  is  divided, 
that  is, 

/»ir/2  /<»2Jr 

Xi  =  p  I        sin  0  cos  OdQ  I      R^  ^\d^  or,  since 

*/0  c/0 


*See  Routh's  Analytical  Statics,  Vol.  II,  §§  182-221.  Tarleton's 
Mathematical  Theory  of  Attraction,  §§  21-24  and  82-105.  Schell's 
Theorie  der  Bewegung  und  der  Krafte,  pp.  690-716. 


120  SURFACE   DISTRIBUTIONS. 

2  abW  cos  0 


b2c2  cos2  6  4-  «2c2  sin2  (9  cos2  <£  4-  a262  sin2  0  sin2 
2  a&V2  cos  0 


u  --  v 


u  +  v  cos2  </>  4-  w  sin2 « 
Now 


u  4-  w  cos2  </>  +  w  sin2  <£ 

/•»  d(f/ 

i     ~ ~ — ) 


2  V(M  +  v)  (u  +  w) 
Hence, 

X,  =  4  aftc.p  f7--  sinflco 

*/o       V  (^2  cos2  0  +  a2  sin2  0)  (c2  cos2  0  +  a2  sin2  0) 
or,  if  s  =  a2  tan2  0, 

X1  =  2a2^7rp  f"  -  —  _ 

pJo  («  -f  «^§/1(«  +  *!>l/*(«  +  Ol/ 

and 


[159] 


?/0Z0f, 


7TO  ^0    I        - 

Jo      (s  +  a2)1/2 


=  2  abc7rpz0M0  =  zQMQ'. 

At  the  positive  ends  of  the  axes  of  the  ellipsoid  the  force 
components  are  -  Xv  -  Ylt  -  Zlf  If  the  ellipsoid  were  made 
of  matter  of  density  p,  repelling  according  to  the  "  Law  of 
Nature,"  the  force  components  at  the  positive  ends  of  the 
axes  would  be  +  Xlt  +  ¥lf  +  Zv 


GREEN'S  THEOREM.  121 


f  (#u  2/D  ^i)  is  a  point  on  the  ellipsoid  —  +  75  +  "2  =  1> 

tt  0  C 

!,  A.f/1,  X^i)  is  a  corresponding  point  on  the  similar  ellipsoid 
?/2         3,2 

2/2  +  ~T~2  =  1>  an(^  tne  straight  line  which  joins  these 
A  o        \  c 

two  points  passes  through  the  origin. 

It  is  to  be  noticed  that  7f0',  Z0f,  MJ  have  the  same  values 
for  all  similar  ellipsoids,  no  matter  what  their  actual  dimen 
sions  may  be,  and  that  the  components  of  the  attraction  at 
corresponding  points  on  two  similar  homogeneous  ellipsoids 
of  equal  density  p  are  to  each  other  as  the  linear  dimensions  of 
the  ellipsoid. 

Since  the  attraction  of  a  homogeneous  ellipsoidal  homoeoid  is 
zero  (Section  12)  at  all  inside  points,  we  may  draw  through  any 
point  P  within  a  homogeneous  ellipsoid  bounded  by  a  surface 
SQJ  a  surface  S,  concentric  with  S0  and  similar  and  similarly 
placed,  and  affirm  that  the  attraction  at  P  is  equal  to  the 
attraction  of  so  much  of  the  whole  ellipsoid  as  lies  within  S. 
If  OP  cuts  S0  in  jP0,  the  attraction  components  at  P  are 


JT=  —  2  abcTrpxK0)  Y=  —  2  abc-KpyL^  Z  —  —  2  abc 
or  -  xK0',  -  yL0',  -  z3I0'  ; 

therefore,  the  resultant  attractions  at  internal  points  on  any 
straight  line  drawn  through  the  centre  of  a  homogeneous  ellip 
soid  are  parallel  in  direction.  They  are  proportional  in  inten 
sity  to  the  distances  of  the  points  from  the  centre. 

The  potential  function  V  within  a  homogeneous  ellipsoid 

of  density  p  bounded  by  'the  surface  '—  +  7-,  +  —,  ,  =  1  is  such 

a2       b'2       <•- 

that  its  derivatives  with  respect  to  x,  y,  and  z  are  respectively 
equal  to  —  2  abcirpxK^  —2abcTTpyLQ1  —2abcTrpz3I^  where 
A"0,  Lw  J/0  have  the  same  values  at  every  point  of  the  solid, 
so  that 

V  =  alcirp(GQ  -  A>2  -  Lvy2  -  J/^-), 


in  which   G0  is  a  constant  to  be  determined  by  computing 
abcTrpG0,  the  value  of  the  potential  function  at  the  centre. 


122  SURFACE   DISTRIBUTIONS. 

The  polar  equation  of  an  ellipsoidal  surface  of  semiaxes  a,  b,  c, 
when  the  origin  is  at  the  centre  and  6  is  the  angle  which  any 
radius  vector  through  the  origin  makes  with  the  a  axis,  is 

,2  =  _  ?w  _ 

b2c*  cos2  B  +  a2c2  sin2  $  cos2  <£  +  a2b2  sin2  6  sin2  < 


u  4-  v  cos2  </>  H-  w  sin2  <£ 

nZn    s+r^ 
j     rBi 
t/o 

/»7r/2  /**»•/ 

I      sin  (9^  I 

«/o  */o 


Using  the  method  of  reduction  already  employed  in  finding 
the  value  of  X^  we  learn  that 


G0  is  an  elliptic  integral  of  the  first  kind,  KM  Lw  and  J/0  are 
elliptical  integrals  of  the  second  kind.     If  a  >  b  >  c, 

(5  +  a2)  >  (5  +  ft2)  >  (5  +  c2)  and  ^ro  <  £0  <  M0, 
and,  unless  5  is  zero, 

(5  +  a2)/(s  4-  #2)  <  a2/62  and  (5  +  62)/(s  4  c2)  <  62/c2. 
The  equation  for  V  may  be  written  in  the  form 

if  z* 

~~''  ' 


-rrp 


so  that  the  equipotential  surfaces  within  a  homogeneous  ellip 
soid  are  a  set  of  ellipsoidal  surfaces  coaxial  with  the  given 
ellipsoid  and  similar  to  each  other.  The  axes  are  in  the 
same  order  of  length  as  are  those  of  the  ellipsoidal  mass,  but 
are  more  nearly  equal.  The  outer  surface  of  the  attracting 
ellipsoid  is  not  equipotential. 

The  differential  equations  of  the  lines  of  force  within  a 
homogeneous  ellipsoid  are  evidently 


GREEN'S  THEOREM.  123 


dx/xK0  = 
so  that  if  the  reciprocals  of  7f0,  Lm  M0  are  represented  by  k,  I,  m, 


The  two  ellipsoidal  surfaces 


are  confocal  if  a'2  =  a2  +  A,  6"  -  b°  +  A,  c'2  =  c2  +  A.  We  will 
assume  for  convenience  that  A  is  positive.  A  point  P'  on 
the  second  surface  S'  is  said  to  correspond  to  a  point  P  on 
the  first  surface  S,  if  #'  :  x  =  a'  :  a,  y'  :  y  =  b'  :  b,  z'  :  z  =  c'  :  c. 

If  Pj  and  P2  are  any  two  points  on  S,  and  P/,  P2'  the 
corresponding  points  on  S',  the  distance  PiP2'  is  equal  to  the 
distance  P/Pa  [Ivory's  Theorem],  as  may  be  seen  by  substi 
tuting  for  a',  b',  and  c'  in  the  following  equation  their  values 
in  terms  of  a,  b,  and  c. 


To  the  points  on  a  chord  EF  of  £,  drawn  parallel  to  the 
x  axis,  correspond  the  points  on  a  parallel  chord  E'F'  of  5'. 
The  lengths  of  these  two  chords  are  as  a  to  a'.  To  the  points 
in  a  slender  prism  Q,  of  cross-section  AyAz,  within  /S,  one 
edge  of  which  is  the  line  EF,  correspond  the  points  in  a 
slender  prism  Q',  of  cross-section  Ay'Az',  or  Ay  •  &z-b'c'/bc, 
within  S',  and  one  edge  of  this  is  the  line  E'F'. 

If  Q  and  Q'  are  made  of  homogeneous  matter  of  equal  den 
sity,  the  x  component  of  the  attraction  at  any  point  P',  on 
the  larger  ellipsoid  S'.  due  to  Q,  is  [Section  6]  equal  to 


124  SURFACE    DISTRIBUTIONS. 

and  the  x  component  of  the  attraction  due  to  Q'  at  the  point 
P  on  S  corresponding  to  P'  is 

pky&zb'c'  f    1  1 

~~bc         \PE~'  ~  P> 

The  quantities  in  the  parentheses  are  equal,  by  Ivory's  Theo 
rem,  and  the  two  attraction  components  are  to  each  other  as 
bc:b'c'.  If  the  whole  space  inside  S1  is  filled  with  homoge 
neous  matter  of  density  p,  the  x  component  at  any  point  P', 
on  S',  of  the  attraction  of  so  much  of  the  mass  as  lies  within  8 

be 

is  equal  to  the  product  of  —  and  the  x  component  of  the 

b  c 

attraction  of  the  whole  mass  at  the  inside  point  P  which  lies 
on  S  and  corresponds  to  P'.  We  have  already  found  an 
expression  for  the  last-named  force  component. 

To  find,  then,  the  attraction  at  the  outside  point  P'  (x',  ?/',  2'), 
due  to  a  homogeneous  ellipsoid  of  density  p  bounded  by  the 

surface  S,  or  -^  +  y^H — 2  =  1>  we  mus^  nrs^  find  the  positive 

a  x'2  ?/'2  z1'2 

value  of  A  which  satisfies  the  cubic  —    -  +  •— (-  — ^  —  =  1. 

a2  +  A       b2  4-  A       c2  4-  A 

and  thus  determine  the  axes  of  the  ellipsoidal  surface  S' 
through  P'  confocal  with  S.  If  we  call  this  value  of  A,  A',  the 
point  P  on  S  which  corresponds  to  P'  on  S'  has  the  coordi- 

/       ax'  by'  CK'      \ 

nates  (  —  ?    —  . . >  _  )>  and  the  x  component 


of  the  attraction  at  P  due  to  an  ellipsoid  of  density  p  bounded 
by  /S"  would  be 

ds 


(s  +  a'2)3/2(S  +  ft'2)1/2^  +  c'2)1/2 

If  we  multiply  this  result  by  bc/b'c',  we  shall  get  the  result 
sought.  If  we  substitute  s  +  A  for  s  in  the  integral  and 
remember  that  x  :  x'  —  a  :  a',  we  may  write  the  x  component 
of  the  attraction  of  the  ellipsoid  at  the  point  P  in  the  form 

ds 


'  f 

»A' 


GREEN'S  THEOKEM.  125 

where  m  is  the  mass  of  the  ellipsoid.     The  components  parallel 
to  the  axes  of  y  and  z  of  the  attraction  at  P'  are,  similarly, 

ds 


_ 


Z  =  -  %mz'j 


(s  +  6 
Lj 
ds 


(s  4-  a2)1/2  (s  4  62)1/2(s  4  c)3/2 
=  -  2  abcvpz'M  =  -  |  w*'  J/. 


We  know  that,  if  we  substitute  in  the  equation 


the  coordinates  of  any  point  in  space,  the  largest  root  of  the 
equation  corresponds  to  an  ellipsoid  passing  through  the  point, 
and  is  negative,  zero,  or  positive  according  as  the  point  lies 
within,  on,  or  without  S.  Following  Dirichlet,  let  us  imagine 
a  function  u  of  the  space  coordinates,  which  shall  have  the 
value  zero  at  every  point  within  or  on  S,  and,  at  every  point 
outside  of  S,  shall  be  equal  to  the  positive  root  of  the  equa 
tion  .F(A)  —  0  which  belongs  to  that  point  ;  and  let  us  con 
sider  the  integral 

jr  i.        fY-i  x*  l?  ^     >\ 

V=  irabcp  I        1  --  -  --  *•  ---  1 

PJu    \         s  +  a*       s  +  b*       8  +  f/ 

ds 


(s  +  a*)1'*  (s  +  62)1/2(s  +  c2)1/2 

which  evidently  vanishes  at  infinity.  For  inside  points  where 
u  is  zero,  V  is  identical  with  the  value  just  found  for  the 
potential  function  within  a  homogeneous  ellipsoid  of  density  p. 
Since  V  involves  x  explicitly  and  also  implicitly  through  u, 
we  have,  in  general,  at  any  outside  point, 


-r\    y  __   _   Q  7 

Cf)JC 


TrabcpDxu  /  x'2  y2  z'2 


«*+*     M-hu 


126  SURFACE    DISTRIBUTIONS. 

but,  from  the  definition  of  u,  the  coefficient  of  Dxu  vanishes 
when  u  is  positive,  so  that  the  integral  alone  remains  and  gives 
the  value  already  found  for  the  x  component  of  the  attraction 
at  an  outside  point  due  to  a  homogeneous  ellipsoid  of  density 
p  bounded  by  S.  At  S,  Dx  V  is  continuous  :  V  gives  every 
where,  therefore,  the  value  of  the  potential  function  due  to  a 
homogeneous  ellipsoid  of  density  p  bounded  by  S. 
If  we  note  that 


r(^  +  _j_  +  ^_\ 

J    \s  -f-  a2       s  4-  b'2      s  +  c2/  (s  +  a 


ds 


*)1'*  (s  +  62)1/2(s  +  C2)1'2 
-2 


(s  -f  a2)l/2(s  -f  P)1/f(i  +  c2)1' 
and  that  the  equation  F(u)  =  0  yields 

2  x         (       x'2  ?/2 

D*U  =  (a2  +  u)  /  \(a*  +  w)2  +  (//2  +  w)2  + 
(with  similar  values  for  Dtf?«  and  Dzii),  so  that 


, 


for  an  outside  point,  and  zero  for  a  point  within  S,  it  is  easy 
to  see  that  V  satisfies  *  Laplace's  Equation  without  S  and 
Poisson's  Equation  within  /S,  as  it  should. 

52.  Logarithmic  Potential  Functions.  When  a  distribution 
of  matter  attracting  or  repelling  according  to  the  "Law  of 
Nature  "  is  such  that  by  a  proper  choice  of  axes  of  reference 
for  a  set  of  orthogonal  Cartesian  coordinates  the  density  can 
be  made  to  depend  on  two  of  these  coordinates  only,  the  dis 
tribution  evidently  extends  indefinitely  far  in  both  directions 
parallel  to  the  third  axis.  Such  a  distribution  is  sometimes 
said  to  be  "columnar."  Any  infinitely  long  cylinder  the 
density  of  every  filament  of  which  is  the  same  throughout 
the  whole  length  of  that  filament,  though  different  filaments 

*  Picard,  Traitt  ft  Analyse,  Vol.  I,  p.  177. 


GREEN'S  THEOREM. 


127 


may  have  different  densities,  is  a  columnar  distribution.  If 
we  choose  for  z  axis  a  line  parallel  to  these  filaments,  the 
components  of  the  force  taken  parallel  to  the  x  and  y  axes  at 
any  point  involve  x  and  y  only,  and  there  is  no  force  compo 
nent  parallel  to  the  axis  of  z.  Since  the  z  coordinate  will  not 
appear  in  any  of  our  equations,  we  may  represent  a  columnar 
distribution  by  its  trace  in  the  xy  plane,  if  we  keep  in  mind 
the  fact  that  the  distribution  itself  extends  to  infinity  in  both 
directions  perpendicular  to  this  plane. 

It  is  evident  from  the  work  of  Section  6  that  a  fine,  homo 
geneous  filament  of  cross-section  A^1?  made  of  repelling  matter 


FIG.  30. 


of  density  plt  urges  a  unit  mass  at  a  point  at  a  distance  r  from 


the  filament  with  a  force  of 


fy        \  A 

:L^  — 


absolute  kinetic  force 


units.  It  follows  that  if  the  trace  of  a  columnar  distribution 
in  the  xy  plane  is  an  area  A^  the  force  components  at  the 
point  (x,  ?/,  z)  parallel  to  the  axes  of  x  and  y  are 

2pl(x-xl)dAl       v       CC   2PiG/-  Vi)dAi 


'"ff^ 

«y  «y    \xi 

where  p^  is  the  density  at  any  point  the  x  and  y  coordinates  of 
which  are  xl  and  yl  respectively,  and  where  the  integrals  are 
to  be  extended  over  the  whole  of  Alf  The  integral 

v  =  +ffpi  ]°g  [(*i  -  *Y  +  (^i  -  y)2] dAi  =ff2  PI  los  ^^i 


128  SURFACE    DISTRIBUTIONS. 

extended  over  A  is  called   from  its  form  the  "logarithmic 
potential  function  "  belonging  to  the  distribution,  and 
X=  +  DXV,    Y=  +  I)yV. 

In  the  general  case  the  columnar  distribution  must  be  con 
sidered  to  be  made  up  partly  of  filaments  of  positive  matter 
and  partly  of  filaments  of  negative  matter,  so  that  the  density 
is  positive  for  some  values  of  x  and  y  and  negative  for  others. 
Under  these  circumstances  X  and  Y  represent  the  force  com 
ponents  which  would  act  on  a  unit  quantity  of  positive  matter 
concentrated  at  the  point  (a?,  y,  z).  It  will  be  convenient  to 
denote  the  amount  of  matter  (reckoned  algebraically)  in  the 
unit  length  of  a  columnar  distribution  by  M.  It  is  evident 
that  at  an  infinite  distance  (in  the  xy  plane)  from  the  trace  Al 
of  a  columnar  distribution  the  logarithmic  potential  function 
becomes  infinite,  unless  M  is  zero,  while  the  force  components 
vanish  in  any  case. 

It  is  easy  to  prove  that,  if  M  is  zero,  V  so  vanishes  at 
infinity  in  the  xy  plane  that,  if  r  is  the  distance  from  any 
finite  point  in  the  plane,  r  V  and  r2Dr  V  have  finite  limits  as 
r  increases  indefinitely.  If  M  is  not  zero,  V  becomes  infinite 
at  infinity  in  such  a  way  that  the  quantities  (  V  —  2  M  log  r), 
(r.DrV-2M),(r.  V-DrV-±  Jf'logr),  and  (V-r>\ogr'DrV) 
all  approach  the  limit  zero  when  r  becomes  infinite.  That 
X,  Y,  and  V  are  finite  at  every  finite  point  in  the  xy  plane 
outside  of  Al  is  evident ;  that  no  one  of  them  is  infinite  at  any 
point  within  Al  can  be  proved  by  transforming  the  integrals 
which  define  them  to  polar  coordinates,  using  the  suspected 
point  as  origin. 

If  n  is  the  exterior  normal  of  any  closed  curve  s  in  the  xy 
plane,  and  r  the  distance  from  any  fixed  point  0  in  the  plane, 
the  line  integral  of  cos  (n,  r)/r  taken  around  s  is  equal  to 
zero,  TT,  or  2  ?r,  according  as  0  is  without,  on,  or  within  s. 
From  this  it  follows  that  the  line  integral  around  any  closed 
curve  in  the  xy  plane,  of  the  normal  outward  component  of 
the  force  due  to  any  columnar  distribution  of  repelling  matter 
the  lines  of  which  are  perpendicular  to  that  plane,  is  equal 


GREEN'S  THEOREM.  129 

to  4?r  times  the  mass  of  the  unit  length  of  so  much  of  the 
columnar  distribution  as  is  surrounded  by  the  curve.  AVc 
may  regard  this  as  Gauss's  Theorem  applied  to  columnar 
distributions. 

If  a  function  u  involves  x  and  y  and  does  not  involve  z,  no 
confusion  need  be  caused  by  denoting  Dx-u  +  D£u  by  \"2n. 
Using  this  notation,  Green's  Theorem  for  functions  of  the  two 
variables  x  and  y  may  be  written  in  the  form 


>J*  '  D*u<  +  D,u  •  Dyiv)  dA 
=  Cu  •  DHw  -ds-  C  Cu  •  V2w  •  dA 
=  Cw  •  Dnu  -ds  —  C  Cw  •  V-u  •  dA, 


where  the  line  integrals  are  to  be  extended  around  a  closed 
curve  s  in  the  xy  plane,  within  and  on  which  u  and  w  with 
their  first  derivatives  are  continuous,  and  the  double  integrals 
extended  over  the  area  shut  in  by  s.  If  in  this  equation  we 
make  u  =  1  and  w  the  logarithmic  potential  function  V,  due 
to  a  columnar  distribution,  we  get 


and  this,  according  to  the  special  form  of  Gauss's  Theorem, 
just  stated,  is  equal  to 


Since  the  form  of  the  curve  *  may  be  chosen  at  pleasure,  it 
must  be  true  that  at  every  point  V'2  V  =  +  4irp.  It  is  desirable 
to  notice  that  the  plus  sign  here  precedes  4  -n-p,  whereas  in  Pois- 
son's  Equation,  as  applied  to  the  Newtonian  Potential  Func 
tion  of  a  finite  mass,  the  corresponding  sign  is  minus.  This 
and  many  other  differences  of  sign  that  appear  in  our  equa 
tions  might  have  been  removed  if  the  opposite  sign  had  been 
given  to  the  integral  which  defines  the  logarithmic  potential 


130  SURFACE    DISTRIBUTIONS. 

of  a  columnar  distribution,  but  if  this  had  been  done  a 
positive  mass  would  have  given  rise  to  a  negative  potential 
function,  and  this  might  have  caused  confusion. 

If  a  portion  of  a  columnar  distribution  consists  of  a  surface 
charge  on  a  cylindrical  surface,  we  may  conveniently  construct 
a  small  quadrilateral  in  the  xy  plane  by  drawing  two  normals 
across  the  ends  of  an  element  of  the  trace  of  the  cylindrical 
surface  and  two  very  near  curves  parallel  to  the  trace  element, 
one  on  one  side  and  the  other  on  the  other.  If,  then,  we  apply 
Gauss's  Theorem  to  this  quadrilateral,  we  shall  learn  that  at 
every  point  of  the  trace  the  sum  of  the  normal  derivatives  of 
V  taken  away  from  the  curve  on  each  side  is  47r<r. 

If  a  closed  curve  s  be  drawn  in  the  xy  plane  so  as  to 
include  the  trace  of  a  portion  of  a  columnar  distribution  the 
lines  of  which  are  perpendicular  to  that  plane  and  to  exclude 
the  trace  of  another  portion,  and  if  Fi  and  Vz  represent  the 
parts  of  the  potential  function  V  belonging  to  these  two 
portions  of  the  distribution,  we  may  apply  Green's  Theorem 
to  V  and  the  logarithm  of  the  distance  from  a  fixed  point  0 
in  the  plane.  If  n  represents  a  normal  pointing  outward  from 
s,  we  shall  find  that 

r) 

' 


/r    •  COS  (71,    i)  (*  T\    Tr    ~\  ^ 
» *  ds  —  I  Dn  V  •  log  r  •  ds 

is  equal  to  the  value  at  0  of  2  TT  F2,  if  0  is  within  s ;  and  to 
the  value  at  0  of  —  2  ?r  F1?  if  0  is  without  5. 

If  s  happens  to  be  a  curve  on  which  F  is  constant, 

log  r  •  ds 


is  equal  to  the  value  at  0  of  FI,  if  0  is  without  s,  or  of  V,  —  F2, 
if  0  is  within  s.  The  reader  may  compare  these  results  with 
those  given  in  equations  [153]  and  [157]. 

If  a  function  w  =f(x,  y)  has  the  value  zero  at  every  point  of 
a  closed  curve  st  in  the  xy  plane  and  the  constant  value  C 
all  over  another  closed  curve  sa,  shut  in  by  slf  and  if  between 
sl  and  s2?  w  is  everywhere  harmonic,  we  may  apply  Green's 


GREEN'S  THEOREM.  131 

Theorem  to  w  and  the  logarithm  of  the  distance  from  a  fixed 
point  0  in  the  plane  and  prove  that 


•where  the  normals  point  outward  on  st  and  inward  on  s2,  is 
equal  to  0,  the  value  of  w  at  0,  or  (7,  according  as  0  is  without 
«!,  between  sl  and  «2,  or  within  s2.  Surface  charges,  of  density 

.   "    ?  applied  to  Sj  and  s2  would,  therefore,  give  rise  to  the 

potential  function  w  between  sl  and  s2. 

If  a  function  ?r  =f(x,  y),  harmonic  at  all  finite  points,  has 
the  constant  value  c  on  a  closed  curve  s  in  the  xy  plane  and 
becomes  infinite  at  infinity  in  this  plane  in  such  a  way  that 

limit  (w  —  2  fi  log  r)  =  0,  or  limit  (/•  log  r  •  Drw  —  w)  =  0, 

where  /A  is  a  given  constant,  then  at  all  points  without  s,  if  n 
is  an  interior  normal, 


and  w  is  the  potential  function  due  to  a  columnar  distribution 
of  superficial  density  —DHw/4:v  on  the  cylindrical  surface 
of  which  s  is  the  right  section.  The  amount  of  matter  in  the 
unit  length  of  this  cylindrical  distribution  is  /u. 

If  within  the  closed  curve  s  in  the  xy  plane,  w  =f(x,  y)  is 
harmonic,  we  may  apply  Green's  Theorem  to  w  and  the  loga 
rithm  of  the  distance  r  from  a  fixed  point  01?  within  s,  using 
as  field  the  region  within  5  and  without  a  small  circumference 
drawn  around  0lP  This  yields 

2  'jrw    (   =        w' 


at  (\  =  \ 

where  n  is  the  exterior  normal  to  s.    If  r.2  is  the  distance  from 
a  fixed  point  0.2,  without  s,  we  may  prove,  in  a  similar  way  that 

0  =  C[w-  Dn  log  r.2  -  log  r2  •  Dnw~]  ds, 


132  SURFACE    DISTRIBUTIONS. 


or 


If  s  is  a  circumference  of  radius  a  with  centre  at  (7,  and  if 
Ol  and  02  are  inverse  points  such,  that 

<70!  =  ll}   C02  =  l>,   IJ*  =  a?, 
then  r1/r2  is  constant  all  over  s, 

f  Dnwds  =  )    )  V*wdxdy  =  0, 

and  2  TTW  at  Oj  =  J  w  [Z>n  log  ^  -  Z>H  log  r2]  ds. 

Moreover  ^  •  I>n  log  r^  =  cos  (T-J,  ri),  r.2  •  J)n  log  ?-2  =  cos  (r2,  w), 
If  =  a2  +  rj2  —  2  a?1!  cos  (ra,  ^), 
Z22  =  a2  +  r22  —  2  ar2  cos  (r2,  n), 
and  the  value  on  s  of  n/r2  is  h/a,  so  that 


taken  around  the  circumference. 

If  we  introduce  polar  coordinates  with  origin  at  the  centre 
of  s  and  denote  the  coordinates  of  0X  by  Zx  and  <£1?  we  shall  have 


__ 
ato^  2        2--' 


This  is  sometimes  called  "Poisson's  Integral.  7? 
At  the  centre  of  the  circumference  where  ^  =  0, 


w 


= f  wds. 

ZiraJ 


EXAMPLES. 


1.  If  the  potential  function  due  to  a  certain  distribution  of 
matter  is  given  equal  to  zero  for  all  space  external  to  a  given 
closed  surface  S  and  equal  to  <f>  (x,  y,  z),  where  <£  is  a  continu 
ous  single-valued  function  (zero  at  all  points  of  $),  in  all  space 


GREEN'S  THEOREM.  133 

within  S'j   there  is  no  matter  without  S,  there  is  a  superficial 
distribution  of  surface  density 


upon  S,  and  the  volume  density  of  the  matter  within  S  is 

P  =  ~  ^  W*  +  D*<t>  +  Z>,*£|. 

[Thomson  and  Tait.] 

2.  Show  that,  if  w  is  constant  on  the  closed  surface  S  and 
is  harmonic  within  S,  it  is  constant  in  the  space  enclosed  by 
S-,  and  that  if  W  vanishes  at  infinity  and  is  everywhere  har 
monic,  it  is  everywhere  equal  to  zero. 

3.  If  two  functions,  wt  and  w2,  which  without  a  closed  sur 
face  S  are  harmonic  and  vanish  at  infinity,  have  on  S  values 
which  at  every  point  are  in  the  ratio  of  X  to  1,  \  being  a  con 
stant,  then  everywhere  u\  =  \u<2. 

4.  The  functions  u  and  v  have  the  constant  values  uv  and  \\ 
on  the  closed  surface  Sl  and  the  constant  values  u.2  and  v2  on 
the  closed  surface  £3  within  Sr     Between  Sl  and  S.2,  u  and  v 
are  harmonic.     Show  that 

(u  -  M!)  (v,  -  vj  =  (u  -  Vl)  (u.2  -  MI). 

5.  Outside  a  closed  surface  S,  u\  and  w.2  are  harmonic  and 
have  the  same  level  surfaces.     wl  vanishes  at  infinity,  while 
w.2  has  everywhere  at  infinity  the  constant  value  C.     Assum 
ing  that  a  scalar  point  function  v  is  expressible  in  terms  of 
another,  u,  if,  and  only  if, 

Dxv/Dxu  =  Dyv/Dyu  =  Dzr/Dzu, 
show  that  u\2  is  of  the  form  Bu\  4-  C. 

6.  Show  that  there  cannot  be  two  different  functions,  W 
and  JF',  both  of  which  within  the  space  enclosed  by  a  given 
surface  S  (1)  satisfy  Laplace's  Equation,   (2)  are,  together 
with  their  first  space  derivatives,   continuous,   and  (3)   are 
either  equal  at  every  point  of  S,  or  satisfy  on  S  the  equation 
DnW=DnW,  and  are  equal  at  some  one  point. 


134  SURFACE   DISTRIBUTIONS. 

7.  Show  that,  given   a  set  of  closed  mutually  exclusive 
surfaces,  there  cannot  be  two  different  functions,  W  and  W, 
which  without  these  surfaces  (1)  satisfy  Laplace's  Equation, 
(2)  are,  with  their  first  space  derivatives,  continuous,  (3)  so 
vanish  at  infinity  that  rW,  rW,  r*DrW,  r*DrW,  where  r  is 
the  distance  from  any  finite  fixed  point,  have  finite  limits,  and 
which  satisfy  one  of  the  following  relations  :    (1)  at  every 
point  on  the  given  surfaces   W  =  W,  (2)  at  every  point  of 
every  surface  DnW  =  DnW, 

8.  At  every  point  of  a  portion  (or  the  whole)  of  a  closed 
surface  S  (or  of  a  set  of  closed  surfaces)  the  functions  wl  and 
w2  have  equal  values,  and  at  every  point  of  the  remainder  of 
S  these  functions  have  equal  normal  derivatives.      Outside 
and  on  S  both  functions  are  harmonic,  and  they  both  vanish 
at  infinity  in  some  manner  not  more  closely  defined.     Each  of 

the  integrals   J  D^dS,    \  Dniv2dS  has  evidently  the  same 

finite  numerical  value  when  taken  over  S  or  over  any  other 
surface  which  encloses  S.  Show  that  wl  and  w2  are  identical. 
If  the  values  of  w^  and  iv.2  at  a  point  P,  the  coordinates 
of  which  referred  to  any  fixed  point  as  origin  are  (r,  0,  <£), 
instead  of  approaching  zero  as  r  is  made  to  increase  indefi 
nitely,  both  approach  the  limit  f(6,  <£),  /  being  a  continuous 
function,  when,  with  any  values  of  0  and  <£,  r  is  made  infinite, 
wl  and  w2  are  identical. 

9.  The  given  closed  surface  Sl  shuts  in  the  given  closed 
surface  S2.      The  given  function  w  is  harmonic  between  Sv 
and  S2.     Show  that  no  other  function  than  w,  harmonic  be 
tween  Sl  and  &,  has  the  same  value  that  w  has  at  every  point 
of  Si  and  the  same  value  of  the  normal  derivative  at  every 
point  of  S2.     Show  also  that  any  such  function  which  has  the 
same  value  of  the  normal  derivative  at  every  point  of  Si  and  S3 
that  the  normal  derivative  of  w  has  differs  from  w  at  most  by 
a  constant.     No  other  function  than  w^  harmonic  between  Si 
and  $2,  has  the  same  value  that  w  has  at  every  point  of  $2 
the  same  value  of  the  normal  derivative  at  every  point  of 


GREEN'S  THEOREM.  135 

10.  The  harmonic  function  <r.  which  BO  vanishes  at  infinity 
that,  if  r  is  the  distance  from  any  fixed  finite  point,  the  limits 
of  r  r  and  rDr  V  are  not  infinite,  has  an  open  zero  level  sur 
face  Si  as  well  as  a  series  of  closed  level  surfaces  of  which  one 
is  Sf     Show  that  in  the  region  T,  between  Sl  and  .Sr  <r  is 
the  potential  function  due  to  surface  distributions  on  Si  and 
S.  defined  by  the  equation  4  w  =  />„  FT.  where  n  points  out 
of  T.     The  whole  charge  on  the  two  surfaces  is  zero. 

11.  Outside  the  closed  surface  N.  upon  which  its  value  is 
given  at  every  point,  the  function  <r  is  harmonic  except  at 
certain  points,  P^  P+,  Py,  etc.,  where  it  becomes  infinite  in 
such  a  way  that,  if  rt  represents  the  distance  from  P^ 

<r  —  ml/rt  is  harmonic  at  J\. 

where  mt  is  a  constant  belonging  to  the  point  P^  At  infinity 
i.r  vanishes  like  a  Newtonian  potential  function.  Prove  that 
»r  is  unique.  If  <r  is  a  Newtonian  potential  function,  what 
do  you  know  about  the  distribution  which  gives  rise  to  it  ? 

12.  The  functions  V,  JF,  9,  17  with  their  first  space  deriva 
tives  are  continuous,  everywhere  without  a  given  closed  sur 
face  S,  and  they  vanish  at  infinity  like  a  Newtonian  potential 
function  due  to  a  finite  distribution  of  matter,    r  and  JT  have 
the  same  values  at  every  point  of  S,  but  outside  «S\  £  ,  B,  and 
O  satisfy  Laplace's  Equation  and  W  does  not.     The  surface 
integrals  of  the  normal  derivatives  of  9  and  O  taken  over 
•V  are  equal,  but  0  has  the  same  value  all  over  S,  and  Q  a 
continuously  variable  value.     Show  that,  if  the  integrations 
embrace  all  space  outside 

'  r  ~  (/"  ' 

<fff     l>  "•r  +  (JPrir) 


>  !)  '  -    /' 


136  SURFACE   DISTRIBUTIONS. 

Hence,  show  that  the  energy  of  a  given  charge  spread  on  a 
given  surface  S  is  least  when  the  arrangement  is  equipo- 
tential. 

13.  Everywhere  within  the  closed  surface  S  the  two  scalar 
point  functions  V  and  V1  are  continuous  with  their  first  deriv 
atives.  Over  a  given  portion  of  S,  V  and  V  have  equal 
values,  while  over  the  remainder  of  S  both  Dn  V  and  Dn  V  are 
equal  to  zero.  The  vectors  q  and  q'  have  the  components 
\DXV9  \DVV,  \DZV  and  \DXV',  \DyV',  \DZV  respectively, 
where  X  is  a  positive  analytic  scalar  point  function.  Show 
that,  if  q  is  solenoidal  and  q'  is  not  solenoidal,  the  integral 


extended  over  the  whole  space  within  S  is  less  than  the  integral 


extended  over  the  same  region. 

14.  Gravitating  matter  of  given  uniform  density  is  confined 
within  a  given  closed  surface,  but  its  volume  is  less  than  that 
enclosed  by  the  surface.     Prove  that  its  potential  energy  is  a 
maximum,  if  the  matter  forms  a  shell  of  which  the  given  sur 
face  is  the  outer  boundary,  while  the  internal  boundary  is  an 
equipotential  surface. 

15.  Let  £  =  /!  (x,  y)   and  y  =  f2  (x,  y)  be  two  analytical 
functions  of  x  and  y  such  that  the  two  families  of  curves 


are  orthogonal.  Let  V  be  any  function  of  x  and  y  which, 
with  its  first  space  derivatives,  is  continuous,  within  and  on  a 
closed  curve  s,  drawn  in  the  coordinate  plane.  Let  h^  and  h^ 
be  the  positive  roots  of  the  equations 

v  -  (•*>.*)'  +  (^vn  v  -  (DM?  +  (z>^)». 

fv\ 

Prove  that  s,  the  surface  integral  of  h%  •  k^-  DA  —  \)  taken  all 


GREEN'S  THEOREM.  137 

over  the  area  enclosed  by  s,  is  equal  to  the  line  integral  taken 
around  s  of  Fcos(£,  ri),  where  n  is  an  exterior  normal  and 
(£,  n)  represents  the  angle  between  n  and  the  direction  in 
which  £  increased  most  rapidly. 

Show  that  the  corresponding  theorem  in  three  dimensions 
may  be  expressed  by  the  equation 

//JV  A  •  D,  (jjjj)  dr  =ff  rcos  (fc  n)d& 

16.  The  operator  [(Z>x)2  +  (£>„)"  +  (A)2]  applied  to  any  of 
the  quantities  x±y±iz  v2,  x±iy  ~v2  ±  «,  etc.,  yields  zero  : 
is  every  analytic  function  of  any  one  of  these   quantities 
harmonic  ? 

17.  The  product  of  two  harmonic  functions,  u,  v,  is  itself 
harmonic  if,  and  only  if,  the  level  surfaces  of  u  and  v  are 
orthogonal.     The  product  of  three  harmonic  functions,  u,  v,  w, 
is  itself  harmonic  if,  and  only  if,  the  level  surfaces  of  u,  v, 
and  ic  are  mutually  orthogonal. 

18.  The  function  w  of  the  two  variables  x  and  y  is  har 
monic  in  the  xy  plane  everywhere  outside  of  the  mutually 
exclusive  closed  curves  sl  and  s2.     Upon  these  curves  w  has 
given  constant  values.      At  infinity,   u-  vanishes   in  such  a 
manner  that,  if  r  is  the  distance  from  any  finite  point  in  the 
xy  plane, 

rhj^  (r  log  r  •  Drw  -  w)  =  0. 

Show  that  w  is  the  potential  function  without  Sj  and  s2,  due 
to  superficial  distributions  defined  by  the  equation  4  TTO-  =  Z^w, 
upon  the  cylindrical  surfaces  of  which  sx  and  s2  are  the  traces. 
In  the  formula  just  given  the  normal  points  outward  at  5X 
and  s2. 

19.  The  function  w  of  the  two  variables  x  and  y  is  har 
monic  everywhere  in  the  xy  plane  except  at  certain  points,  P,, 
P2,  P3,  etc.,  where  it  becomes  infinite  in  such  a  manner  that, 
if  rk  is  the  distance  from  PkJ  u-  —  2  ^k  log  rk  is  harmonic  at  Pk 
where  ^  is  a  constant  belonging  to  Pk.     Upon  a  certain  open 


138  SURFACE   DISTRIBUTIONS. 

« 

curve,  s,  w  has  the  value  zero,  and  everywhere  at  an  infinite 
distance  from  the  origin  w  so  vanishes  that 


show  that  on  either  side  of  s,  w  may  be  considered  as  the 
logarithmic  potential  function  due  to  a  distribution  of  elec 

tricity  of  density  a-  =  —  -f—  on  the  infinite  cylindrical  surface 

4  7T 

of  which  s  is  a  right  section,  and  to  distributions  upon  lines 
normal  to  the  xy  plane  which  cut  the  plane  at  so  many  of  the 
P  points  as  lie  on  the  chosen  side  of  s. 

20.  If  the  normal  component  of  a  vector  is  zero  at  every 
point  of  a  closed  surface  S,  and  if  'within  and  on  $the  vector  is 
everywhere  solenoidal  and  lamellar,  its.  components  are  equal 
to  zero  at  every  point  within  S.     If  the  normal  component  of 
a  vector  is  given  at  every  point  of  S,  and  if  everywhere  within 
S  the  curl  and  the  divergence  have  given  values,  the  vector  is 
determined.     If  q  and  q'  are  vectors  the  normal  components 
of  which  vanish  at  every  point  of  S,  and  if  within  S,  q  is 
solenoidal  with  curl  k,  while  q'  is  lamellar  with  divergence 
Z>,  wheref^)and  D  are  given  scalar  point  functions,  q  +  q'  is 

*7  the  unique  vector,  the  normal  component  of  which  is  zero  at 

every  point  of  S,  and  which  within  S  has  the  curl  k  and  the 
divergence  D. 

21.  The  normal  derivative  of  u  with  respect  to  v  is 

(Dxu  •  Dxv  -f-  Dyu  •  Dyv  +  Dzu  •  Dzv)/h*. 

22.  If  u  =  xyz,  v  =  2  x  +  y  +  z,  the  values  at  (1,  1,  1)  of 
Dvu  and  Duv  are  2/3  and  4/3. 

23.  The  gradients  of  u  and  v  are  numerically  equal  at  every 
point,  though  not  in  general  coincident  in  direction,  if,  and 
only  if,  u  +  v  and  u  —  v  are  orthogonal  functions.     If  the 
gradients  of  u  and  v  agree  everywhere  in  direction'  though 
not  in  magnitude,  v  is  expressible  as  a  function  of  u,  so  that 


GREEN'S  THEOREM.  139 

24.  If  the  components  parallel  to  the  axes  of  x  and  y  of 
the  solenoidal  vector  (if,  v,  0),  which  has  no  component  parallel 
to  the  z  axis,  are  independent  of  z,  a  vector,  directed  parallel 
to  the  z  axis,  which  has  for  its  intensity  any  partial  inte 
gral  (Qz)  of  u  with  respect  to  y  which  satisfies  the  condition 
DXQZ  =  —  v,  is  a  vector  potential  function  of  the  original 
vector.     Thus :  (0,  0,  x*y  +  if  —  z9)  is  a  vector  potential  func 
tion  of  (z2  +  3?/2,  9xs-2xy,  0).     The  value  at  the  point 
(x,  y,  z)  of  the  derivative  of  Qz,  taken  in  a  direction  perpen 
dicular  to  the  z  axis  and  making  an  angle  a  +  90°  with  the 
plane  of  xz,   is  DXQS  •  cos  (a  +  90°)  +  DVQZ  •  sin  (a  +  90°),  or 
r>,,Qz  •  cos  a  —  DXQ.  •  sin  a,  or  u  cos  a  +  v  sin  a,  and  this  is  the 
resolved  part  of  the  vector  .(u,  u,  0)  at  the  same  point  in  a 
direction  parallel  to  the  xy  plane  and  making  an  angle  a 
with  the  plane  of  xz. '    We  learn,  therefore,  that  the  numer 
ical  value  at  any  point   P  of  the  derivative    of    Q^   taken 
in  any  direction  s  parallel  to  the  xy  plane,  is  equal  to  the 
component  of  the  vector  (M,  v,  0)  in  a  direction  parallel  to 
the  xy  plane  and  perpendicular  to  s.     Show  that  the  inter 
section  of  any  plane  parallel  to  the  xy  plane  with  a  cylin 
der   of  the   family    Qz  =  constant   is   a  line   of  the   vector 
(u,  v,  0).      Show  also   that   DX*QZ  +  D*QZ  =  -  (Dxv  -  D,u), 
the  negative  of  the  component  parallel  to  the  «  axis  of  the 
curl  of  (u,  v,  0). 

25.  A  vector  parallel  to  the  x  axis  of  intensity  independ 
ent  of  z  and  equal  to  the  negative  of  a  partial  integral  of  w 
with  respect  to  y,  and  a  vector  parallel  to  the  y  axis  of  inten 
sity  independent  of  z  and  equal  to  a  partial  integral  of  w 
with  respect  to  x,  are  vector  potential  functions  of  the  vector 
(0,  0,  w),  provided  w  is  independent  of  z.     For  example  :  the 
vectors  [y2  -  3x*y  +/(»),  0,  0]  and  [0,  x*  -  2xy  +  <£<»,  0] 
are  vector  potential  functions  of  the  vector  (0,  0,  3x2  —  2y). 

26.  If  the  lines  of  a  vector  are  circles  parallel  to  the  xy 
plane  with  centres  on  the  z  axis,  and  if  the  intensity  of  the 
vector  is  a  function  f(r)  of  the  distance  r  from  that  axis,  a 
vector,  everywhere  parallel  to  the  z  axis,  of  intensity  F(r), 


140  SURFACE   DISTRIBUTIONS. 

where  f(r)  =  —  DrF(r)  is  a  vector  potential  function  of  the 
original  vector.     Is  this  original  vector  solenoidal  ? 

27.  If  the  lines  of  a  vector  are  straight  lines  parallel  to  the 
xy  plane  and  emanating  from  the  z  axis,  and  if  the  intensity 
of  the  vector  is  a  function  /(/•)  of  the  distance  r  from  this 
axis,/(r)  must  be  of  the  form  c/r  if  the  vector  is  solenoidal. 
A  vector  with  such  lines  as  these  cannot  be  solenoidal  if  the 
intensity  at  every  point  is  a  given  function  of  the  angle  which 
the  line  of  the  vector  through  that  point  makes  with  the  xz 
plane. 

28.  The  lines   of   the  vector  [jc  -f(x,  y),  y  -f(x,  y),  0]  are 
straight  lines  parallel  to  the  xy  plane  and  emanating  from 
the  z  axis,  and  its  curl  is  of  the  form  (0,  0,  y  •  Dxf  —  x  -  Dyf). 
If/  is  expressible  as  a  function  of  the  angle  tanr*.(y/a;), 
y  •  Dxf  —  x  •  Dyf  is   also    expressible   as   a  function   of   this 
angle,  but  if  /  is  expressible  as  a  function  of  r  =  Vx2  -f-  y2, 
y  •  Dxf  —  x  •  Dyf  vanishes  and  no  vector  of  the  form 


J 


can  be  a  vector  potential  function  of  the  vector  [0,  0,  <£(>)]• 
If  the  ratio  of  y  to  x  be    denoted  by  //-,  and  if  /(/n)  = 

the   vector    |>  •/(/*),    ?/•/<»,   0]    is    a   vector 
-\-  1 

potential  function  of  the  vector  [0,  0,  <£(/K)]. 

29.  The  lines  of  the  vector  [—  y  -f(x,  y},  x  -f(x,  y),  0]  are 
circles  parallel  to  the  xy  plane  with  centres  on  the  2  axis,  and 
its  curl  is  of  the  form  (0,  0,  2f+x-Dxf+  V  •  Dyf).  Show 
that  if  /  is  expressible  as  a  function  of  r,  the  distance  from 
the  z  axis,  so  is  2f+x-  Dxf  +  y  •  Dyf,  and  that,  if 


\_—y-F(r\  x-F(r),  0]  is  a  vector  potential  function  of  the 
solenoidal  vector  [0,  0,  <£(/•)].  Show  also  that  if  /is  expressi 
ble  as  a  function  of  the  angle  tan"1  (y/x),  that  is,  as  a  function 
of  the  ratio,  ^  of  y  to  x,  2f+  x  •  Dxf+  y  -  Dyf  is  expressible 


GREEN'S  THEOREM.  141 

as  a  function  of  //,,  and  that  [—  %  y  •  /(/"•)>  ~kx  •/(/*)>  0]  is  a 
vector  potential  function  of  [0,  0,  /(//.)]. 

30.  The  difference  between  the  values  at  any  two  points 
.4  and  B  of  any  analytic  scalar  point  function  V  is  equal  to 
the  line  integral  taken  along  any  path  from  A  to  B  of  the 
tangential  component  of  the  vector  (DXV,  DyV<,  D.V). 

31.  The  only  families  of  plane  curves  which  are  at  once 
the  right  sections  of  possible  systems  of  equipotential  cylin 
drical  surfaces  in  empty  space  due  to  columnar  distributions 
of  matter  which  attracts  according  to  the  "  Law  of  Nature," 
and  also  the  generating  curves  of  possible  systems  of  equipo 
tential  surfaces  of  revolution  due  to  distributions   of  such 
matter  symmetrical  about  the  common  axis  of  these  surfaces, 
are  families  of  concentric  conies.     Must  every  such  family  of 
conies  be  confocal  ?  [Am.  Jour.  Math.,  1896.] 

32.  If  a  vector  is  determined  at  every  point  by  means  of  the 
components  (R,  0,  Z)  in  the  directions  in  which  the  columnar 
coordinates  of  the  point  increase  most  rapidly,  the  divergence 
of  the  vector  may  be  written  DrR  +  R/  r  +  D0®/r  +  D.Z. 

33.  The  equation 


represents,  when  a,  b,  and  c  are  fixed,  a  family  of  confocal  quad- 
ric  surfaces  of  which  X  is  the  parameter.  If  a>b>c,  and  if 
x,  y,  and  z  are  chosen  at  pleasure,  the  cubic  in  X  has  three  real 
roots  (it,  v,  w)  ;  one  between  —  a2  and  —  ft2,  corresponding  to 
a  parted  hyperboloid,  one  between  —  b2  and  —  c2,  correspond 
ing  to  an  unparted  hyperboloid,  and  one  between  —  c2  and  oo, 
corresponding  to  an  ellipsoid,  so  that  through  every  point  of 
space  three  surfaces  of  the  family  can  be  drawn,  and  it  is 
easy  to  see  that  these  cut  each  other  orthogonally.  The 
direction  cosines  of  a  surface  of  constant  X  have  the  values 
Dx\/h,  Dy\/h,  Dx\/h,  where  /V2  =  (T^A)2  +  (D^  +  (AX)2. 


Dx  X  =  -  2  x/(a?  +  X)  D^F,  and  ?i2  =  - 


142  SURFACE    DISTRIBUTIONS. 

Belonging  to  every  point  in  space  are  three  values  of  A 
(?/,  v,  w),  and  three  values  of  h  (ku,  hv,  /*„,),  and,  if  we  sub 
stitute  u,  v,  and  w  successively  for  A  in  the  equation  FQC)  =  0, 
we  shall  get  three  linear  equations  in  x2,  y2,  z2  from  which 
we  may  obtain  expressions  for  x,  y,  z  in  terms  of  u,  v,  w. 


h2  =  -  4  [(»2  +  «)  (b2  +  u)  o2  4  M>]  /[(M  -  v)  (u  -  w)]» 

and  hv2  and  hw2  have  corresponding  values  which,  substituted  in 


gives  Laplace's  Equation  in  terms  of  the  orthogonal  curvi 
linear  coordinates  (M,  v,  w).  Prove  that  if  we  assume  that  a 
solution  of  this  equation  exists  which  involves  w  only  and 
vanishes  when  w  is  infinite,  the  equation  which  determines 
this  solution  takes  the  form 

Dw\[(a2  4  w)  (b2  4  iv)  (c2  4  w)]1'2 •  Dw  V \  =  0, 

/rim 
71 
(d 


4  tv)l/2(b2  4  'iv)l/2(c2  4  w)1/a 

=  C  f" ^ 

Jw    (a2  4  *^)1/2(62  4  ^6•)1/2(c2  4  w)1/a 

Hence,  show  that  a  set  of  confocal  ellipsoids  are  possible 
external  equipotential  surfaces,  and  that  if  M  is  the  mass  of 
the  corresponding  distribution  the  potential  function  is  given 
by  the  last  equation,  in  which,  since  a  very  large  value  of  w 
corresponds  to  an  ellipsoid  little  different  from  a  sphere  of 
radius  Vt0,  C  is  to  be  determined  by  the  equation 


Find  the  density  of  a  superficial  distribution  on  a  surface  of 
the  w  family,  the  potential  function  of  which  at  all  outside 
points  shall  be  the  function  just  defined. 


143 

34.  The  curl  of  the  curl  of  a  solenoidal  vector  such  that  the 
three  functions  which  give  the  strengths  of  its  components 
parallel  to  the    coordinate  axes  satisfy    Laplace's    Equation 
vanishes.     If  the  lines  of  a  vector  are  all  parallel  to  a  plane, 
and  the  vector  has  the  same  value  at  all  points  in  any  line  per 
pendicular  to  the  plane,  the  vector  is  perpendicular  to  its  curl. 

35.  A  certain  vector,  the  tensor  of  which  is  f(x,  y,  z),  is  at 
every  point  directed  exactly  in  the  direction  of  the  straight 
line  which  joins  the  origin  with  the  point  in  question ;  show 
that  the  vector  is  not  necessarily  lamellar,  but  that  it  is  per 
pendicular  to  its  curl.     If  all  the  components  of  a  vector  are 
functions  of  x  and  y  only,  or  if  all  are  functions  of  x  only,  or 
if  one  component  vanishes   and   the   other  components   are 
functions    of   x,  y,  and  2,  the  vector    may   or  may  not  be 
perpendicular  to  its  curl. 

36.  If  (Qx,   Qy,   Qz)   are   the    components    of   a  vector  Q, 
(\i,  pi,  v:)  the  curl  components,  (\.2,  /x2,  v2)  the  components 
of  the  curl  of  the  curl  of  Q,  and  so  on, 

At  =  DyQz  -  DzQy,  \,  =  Dx  (Div  Q)  -  V2&, 

X3  =  —  V2Xi,  A.4  —  —  V2A2,  and  so  on. 

How    are    these    equations    changed    if    Q   is    a    solenoidal 
vector  ? 

37.  If  the  harmonic  function  f(x,  y,  z)  represents  the  x 
component  of  a  vector  which  is  both  solenoidal  and  lamellar, 
the  y  and  z  components  must  be  of  the  form 


where  \f/(y,  z)  is  a  solution  of  the  equation 

38.  A  certain  vector  (X,  Y,  Z)  is  not  perpendicular  to  its 
curl  (Kx,  Ky,  Kg).  Show  that  the  scalar  function  F,  deter 
mined  from  the  equation 

7ix  •  DXF  +  Ky  •  DyF  4-  Kz  •  DZF  =  -  (K^  +  Ky  Y 


144  SURFACE   DISTRIBUTIONS. 

is  the  scalar  potential  function  of  a  lamellar  vector  (L,  M,  N), 
which  added  to  the  first  vector  gives  a  new  vector  perpendic 
ular  to  its  curl.  Is  this  equation  always  integrable  ? 

39.  A  vector  Q,  with  components  (Qx,  Qy,  Q^,  is  continuous 
except  at  a  certain  surface  S.     In  each  of  the  regions  sepa- 
rated  by  S,  DxQtJ  =  DyQx,  DXQZ  =  VZQX,  DyQz  =  VzQy,  so  that 
at  every  point  within  these  regions  the  curl  of  Q  vanishes. 
Investigate  the  value  of  the  curl  of  Q  on  S  when  the  normal 
(or  a  tangential)  component  of  Q  is  discontinuous  there. 

40.  Unless  V2f=  0,  a  vector  the  x  component  of  which  is 
f(x,  y,  z)  cannot  be  both  lamellar  and  solenoidal. 

41.  Matter  spread  uniformly  in  a  superficial  distribution 
on  a  circular  portion  of  a  plane  forms  a  "circular  surface 
distribution."     Two  such  distributions,  each  of  radius  a,  are 
placed  parallel  and  opposite  each  other  at  a  distance  8  apart. 
If  the  density  of  one  of  these  be  +  a-  and  that  of  the  other 
—  cr,  and  if  8  be  made  to  approach  zero  and  o-  to  increase  in 
such  a  manner  that  the  product  of  a-  and  8  is  always  equal  to 
the  constant  ^  the  resulting  value  of  the  potential  function 
is  said  to  be  due  to  a  "  circular  double  layer  "  of  radius  a,  and 
density  p.      Show  that  the  limiting  value  of  the  potential 
function  at  a  point  P  on  the  axis  of  the  double  layer  and  at 
a  distance  x  from  its  plane  is  ±  2^^  (I  —  x/^/a2  +  x2),  where 
the  positive  sign  is  to  be  used  if  P  is  on  one  side  of  the  double 
layer,  and  the  negative  sign  if  P  is  on  the  other  side.     Is  the 
potential  function  discontinuous  at  the  double  layer  ?     Is  the 
force  discontinuous  ? 

42.  Assuming  the  surface  of  the  earth  as  defined  by  the  sea- 
level  to  be  a  spheroid  of  ellipticity  e,  prove  that  the  mass  of 
the  earth  in  astronomical  units  is  a0Vo  (1  +  e  —  f  m),  where 
gQ  is  the  force  t>f  gravity  at  the  equator,  a0  the  equatorial 
radius,  and  in  the  ratio  of  "  centrifugal  force  "  to  true  gravity 
at  the  equator. 


ELECTROSTATICS.  145 


CHAPTER   V. 

THE  ELEMENTS  OF  THE  MATHEMATICAL  THEOKY 
OP  ELEOTKIOITY, 

I.     ELECTROSTATICS. 

53.  Introductory.  Having  considered  abstractly  a  few  of 
the  characteristic  properties  of  what  has  been  called  "the  New 
tonian  potential  function,"  we  will  devote  this  chapter  to  a  very 
brief  discussion  of  some  general  principles  of  Electrostatics  and 
Electrokinetics.  By  so  doing  we  shall  incidentally  learn  how 
to  apply  to  the  treatment  of  certain  practical  problems  many  of 
the  theorems  that  we  have  proved  in  the  preceding  chapters. 

In  what  follows,  the  reader  is  supposed  to  be  familiar  with 
such  electrostatic  phenomena  as  are  described  in  the  first  few 
chapters  of  treatises  on  Statical  Electricity,  and  with  the  hypoth 
eses  that  are  given  to  explain  these  phenomena. 

Without  expressing  any  opinion  with  regard  to  the  physical 
nature  of  what  is  called  electrification^  we  shall  here  take  for 
granted  that  whether  it  is  due  to  the  presence  of  some  sub 
stance,  or  is  only  the  consequence  of  a  mode  of  motion  or  of  a 
state  of  polarization,  we  may,  without  error  in  our  results,  use 
some  of  the  language  of  the  old  "Two  Fluid  Theory  of  Elec 
tricity  "  as  the  basis  of  our  mathematical  work. 

The  reader  is  reminded  that,  among  other  things,  this  theory 
teaches  that :  — 

(1)  Every  particle  of  a  body  which  is  in  its  natural  state  con 
tains,  combined  together  so  as  to  cancel  each  other's  effects  at 
all  outside  points,  equal  large  quantities  of  two  kinds  of  elec 
tricity  with  properties  like  those  of  the  positive  and  negative 
"  matter"  described  in  Section  44. 

(2)  Electrification   consists   in   destroying  in  some  way  the 
equality  between  the  amounts  of  the  two  kinds  of  electricity 
which  a  body,  or  some  part  of  a  body,  naturally  contains,  so 
that  there   shall  be  an  excess  or  charge  of  one  kind.     If  the 


146  ELECTROSTATICS. 

charge  is  of  positive  electricity,  the  body  is  said  to  be  posi 
tively  electrified  ;  if  the  charge  is  negative,  negatively  electrified. 
Either  kind  of  electricity  existing  uncombined  with  an  equal 
quantity  of  the  other  kind,  is  called  free  electricity. 

(3)  When  a  charged  body  A  is  brought  into  the  neighborhood 
of  another  body  B  in  its  natural  state,  the  two  kinds  of  elec 
tricity  in  every  particle  of  B  tend  to  separate  from  each  other, 
one  being  attracted  and  the  other  repelled  by  ^L's  charge,  and 
to  move  in  opposite  directions. 

In  general,  a  tendency  to  separation  occurs  in  all  parts  of  the 
body,  whether  it  is  charged  or  not,  where  the  resultant  electric 
force  (the  force  due  to  all  the  free  electricity  in  existence)  is 
not  zero.  This  effect  is  said  to  be  due  to  induction. 

In  our  work  we  shall  assume  all  this  to  be  true,  and  proceed 
to  apply  the  principles  stated  in  Section  44  to  the  treatment  of 
problems  involving  distributions  of  electricity.  We  shall  find  it 
convenient  to  distinguish  between  conductors,  which  offer  prac 
tically  no  resistance  to  the  passage  of  electricity  through  their 
substance,  and  nonconductors,  which  we  shall  regard  as  prevent 
ing  altogether  such  transfer  of  electricity  from  part  to  part. 

54.  The  Charges  on  Conductors  are  Superficial.  When  elec 
tricity  is  communicated  to  a  conductor,  a  state  of  equilibrium  is 
soon  established.  After  this  has  taken  place,  there  can  be  no 
resultant  force  tending  to  move  any  portion  of  the  charge 
through  the  substance  of  the  conductor,  for,  by  supposition,  the 
conductor  does  not  prevent  the  passage  of  electricity  through 
itself. 

Moreover,  the  resultant  electric  force  must  be  zero  at  all 
points  in  the  substance  of  a  conductor  in  electric  equilibrium ; 
for  if  the  force  were  not  zero  at  an}'  point,  electricity  would 
be  produced  by  induction  at  that  point,  and  carried  away 
through  the  body  of  the  conductor  under  the  action  of  the 
inducing  force. 

From  this  it  follows  that  the  potential  function  V,  due  to  all 
the  free  electricity  in  existence,  must  be  constant  throughout 


ELECTROSTATICS.  147 

the  substance  of  any  single  conductor  in  electric  equilibrium, 
whether  or  not  the  conductor  be  charged,  and  whether  or  not 
there  be  other  charged  or  uncharged  conductors  in  the  neigh 
borhood.  Different  conductors  existing  together  will  in  general 
be  at  different  potentials,  but  all  the  points  of  an}'  one  of  these 
conductors  will  be  at  the  same  potential. 

Wherever  V  is  constant,  V2T"=0,  and  hence,  by  Poisson's 
Equation,  p  =  0,  so  that  there  can  be  no  free  electricity  within 
the  substance  of  a  conductor  in  equilibrium,  and  the  whole 
charge  must  be  distributed  upon  the  surface.  Experiment 
shows  that  we  must  regard  the  thickness  of  charges  spread  upon 
conductors  as  inappreciable,  and  that  it  is  best  to  consider  that 
in  such  cases  we  have  to  do  with  really  superficial  distributions 
of  electricity,  in  which  the  conductor  bears  a  rough  analogy  to 
the  cavity  enclosed  by  the  thin  shells  of  repelling  matter  de 
scribed  in  the  preceding  chapter. 

The  surface  density  at  any  point  of  a  superficial  distribution 
of  electricity  shall  be  taken  positive  or  negative,  according  as 
the  electricity  at  that  point  is  positive  or  negative,  and  the  force 
which  would  act  upon  a  unit  of  positive  electricity  if  it  were 
concentrated  at  a  point  P  without  disturbing  existing  distribu 
tions  shall  be  called  ;ithe  electric  force"  or  ••  the  strength  of 
the  electric  field  at  P." 

It  is  evident,  from  Sections  45  and  46,  that  the  electric  force 
at  a  point  just  outside  a  charged  conductor,  at  a  place  where 
the  surface  density  of  the  charge  is  o-,  is  47r<r,  and  that  this  is 
directed  outwards  or  inwards,  according  as  a-  is  positive  or  nega 
tive. 

In  other  words,  DnV,  the  derivative  of  the  potential  function 
in  the  direction  of  the  exterior  normal,  is  equal  to  —  4  TTO-,  and 
the  value  of  V  at  a  point  P  just  outside  the  conductor  is  greater 
or  less  than  its  value  within  the  conductor,  according  as  the 
surface  density  of  the  conductor's  charge  in  the  neighborhood  of 
P  is  negative  or  positive. 

It  is  to  be  carefully  noted  that,  although  the  surface  of  a  con 
ductor  must  always  be  equipoteutial.  the  superficial  density  of 


148  ELECTKOSTATICS. 

the  conductor's  charge  need  not  bo  the  same  at  all  parts  of  the 
surface.  We  shall  soon  meet  with  cases  where  the  electricity 
on  a  conductor's  surface  is  at  some  points  positive  and  at  others 
negative,  and  with  other  cases  where  the  sign  of  the  potential 
function  inside  and  on  a  conductor  is  of  opposite  sign  to  the 
charge . 

It  is  evident,  from  the  work  of  Section  47,  that  the  resistance 
per  unit  of  area  which  the  nonconducting  medium  about  a  con 
ductor  has  to  exert  upon  the  conductor's  charge  to  prevent  it 
from  flying  off,  is,  at  a  part  where  the  density  is  o-,  2rra2. 

55.  General  Principles  which  follow  directly  from  the  Theory 
of  the  Newtonian  Potential  Function.  If  two  different  distribu 
tions  of  electricity,  which  have  the  same  system  of  equipoten- 
tial  surfaces  throughout  a  certain  region,  be  superposed  so  as  to 
exist  together,  the  new  distribution  will  have  the  same  equipo- 
tential  surfaces  in  that  region  as  each  of  the  components.  For, 
if  FI  and  F2,  the  potential  functions  due  to  the  two  components 
respectively,  be  both  constant  over  any  surface,  their  sum  will 
l>e  constant  over  the  same  surface. 

Two  distributions  of  electricity,  which  have  densities  every 
where  equal  in  magnitude  but  opposite  in  sign,  have  the  same 
system  of  equipotential  surfaces,  and,  if  superposed,  have  no 
effect  at  any  point  in  space. 

Two  distributions  of  electricity,  arranged  successively  on  the 
same  conductor  so  that  at  every  point  the  density  of  the  one 
is  m  times  that  of  the  other,  have  the  same  system  of  equipo 
tential  surfaces,  and  the  potential  function  due  to  the  first  is 
everywhere  m  times  as  great  as  that  due  to  the  second. 

If  the  whole  charge  of  a  conductor  which  is  not  exposed  to 
the  action  of  any  electricity  except  its  own  is  zero,  the  super 
ficial  density  must  be  zero  at  all  points  of  the  surface,  and  the 
conductor  is  in  its  natural  state.  For  if  o-  is  not  everywhere 
zero,  it  must  be  in  some  places  positive  and  in  others  negative  ; 
and,  according  to  the  work  of  the  last  section,  the  potential 
function  F,  due  to  this  charge,  must  have,  somewhere  outside 


ELECTKOSTATICS.  149 

the  conductor,  values  higher  and  lower  than  T",,  its  value  in  the 
conductor  itself.  But  this  would  necessitate  somewhere  in  empty 
space  a  value  of  the  potential  function  not  lying  between  V0  and 
0,  the  value  at  infinity  ;  that  is,  a  maximum  in  empty  space  if 
VQ  is  positive,  and  a  minimum  if  T"0  is  negative  ;  which  is 
absurd. 

A  system  of  conductors,  on  each  of  which  the  charge  is  null, 
must  be  in  the  natural  state  if  exposed  to  the  action  of  no  out 
side  electricity.  For,  by  applying  the  reasoning  just  used  to 
that  conductor  in  which  the  potential  function  is  supposed  to 
have  the  value  most  widely  different  from  zero,  we  may  show 
that  the  surface  density  all  over  the  conductor  is  zero,  so  that 
no  influence  is  exercised  on  outside  bodies  ;  and  then,  suppos 
ing  this  conductor  removed,  we  may  proceed  in  the  same  way 
with  the  system  made  up  of  the  remaining  conductors. 

If  a  charge  M  of  electricity,  when  given  to  a  conductor,  ar 
ranges  itself  in  equilibrium  so  as  to  give  the  surface  density 

a-  =/(#,  ;y,  z)  and  to  make  the    potential    function  T\,  =  I  - 

•J       T 

constant  within  the  conductor,  a  charge  —  J/,  if  arranged  on  the 
conductor  so  as  to  give  at  every  point  the  density  —<r=  —f(x,y,z) 
would  be  in  equilibrium,  for  it  would  give  everywhere  the  poten 
tial  function  I  I  ^lf_f  =  _  y^  an(]  this  is  constant  wherever  V^ 

J        r 
is  constant. 

Only  one  distribution  of  the  same  quantity  of  electricity  M  on 
the  same  conductor,  removed  from  the  influence  of  all  other 
electricity,  is  possible  ;  for,  suppose  two  different  values  of  sur 
face  density  possible,  o-j  =fl(x^  y,  z)  and  o-2  =/2  (a1,  y,  z),  then 
—  o-2  =  —  /2(.T,#,  z}  is  a  possible  distribution  of  the  charge  —  J/. 
Superpose  the  distribution  —  a-.2  upon  the  distribution  o-j  so  that 
the  total  charge  shall  be  equal  to  zero ;  then  the  surface  density 
at  every  point  is  a-l—  o-2,  and  this  must  be  zero  by  what  we  have 
just  proved,  so  that  ^  =  o-2. 

Since  we  may  superpose  on  the  same  conductor  a  number  of 
distributions,  each  one  of  which  is  by  itself  in  equilibrium,  it  is 


150  ELECTROSTATICS. 

easy  to  sec  that  if  the  whole  quantity  of  electricity  on  any  con 
ductor  be  changed  in  a  given  ratio,  the  density  at  each  point 
will  be  changed  in  the  same  ratio. 

56.  Tubes  of  Force  and  their  Properties.  We  have  seen  that 
a  unit  of  positive  electricity  concentrated  at  a  point  P  just  out 
side  a  conductor  would  be  urged  away  from  the  conductor  or 
drawn  towards  it,  according  as  that  point  on  the  conductor  which 
is  nearest  P  is  positively  or  negatively  electrified.  If  we  regard 
lines  of  force  drawn  in  an  electric  field  as  generated  by  points 
moving  from  places  of  higher  potential  to  places  of  lower  poten 
tial,  we  may  say  that  a  line  of  force  proceeds  from  every  point 
of  a  conductor  where  the  surface  density  is  positive,  and  that  :i 
line  of  force  ends  at  every  point  of  a  conductor  where  the  sur 
face  density  is  negative.  No  line  of  force  either  leaves  or 
enters  a  conductor  at  a  point  where  the  surface  densit}7  is  zero, 
and  no  line  of  force  can  start  at  one  point  of  a  conductor  where 
the  electrification  is  positive  and  return  to  the  same  conductor 
at  a  point  where  the  electrification  is  negative.  No  line  of  force 
can  proceed  from  one  conductor  at  a  point  electrified  in  an}T  way 
and  enter  another  conductor  at  a  point  where  the  electrification 
has  the  same  name  as  at  the  starting-point.  A  line  of  force 
never  cuts  through  a  conductor  so  as  to  come  out  at  the  other 
side,  for  the  force  at  ever}'  point  inside  a  conductor  is  zero. 

Lines  and  tubes  of  force  are  sometimes  called  in  electrostatics 
lines  and  tubes  of  "  induction." 

When  a  tube  of  force  joins  two  conductors,  the  charges  Q}, 
Q2  of  the  portions  $15  $a  which  it  cuts  from  the  two  surfaces  are 


made  up  of  equal  quantities  of  opposite  kinds  of  electricity. 
For  if  we  suppose  the  tube  of  force  to  be  arbitrarily  prolonged 


ELECTROSTATICS.  151 

and  closed  at  the  ends  inside  the  two  conductors,  the  surface 
integral  of  normal  force  taken  over  the  box  thus  formed  is  zero, 
for  the  part  outside  the  conductors  }'ields  nothing,  since  the  re 
sultant  force  is  tangential  to  it,  and  there  is  no  resultant  force 
at  any  point  inside  a  conductor.  It  follows,  from  Gauss's 
Theorem,  that  the  whole  quantity  of  electricity  (Qi  +  Qi)  inside 
the  box  must  be  zero,  or  Qi  =  —  Q2->  which  proves  the  theorem. 
If  (Tl  and  o-o  are  the  average  values  of  the  surface  densities  of 
the  charges  on  S\  and  S2  respectively,  we  have  <riSi  =  Ql  and 
o-2  So  =  Q2,  whence 

-  =  --,|  [162] 

The  integral  taken  over  any  surface,  closed  or  not,  of  the 
force  normal  to  that  surface  is  called  by  some  writers  the  flow 
of  force  across  the  surface  in  question,  and  by  others  the  induc 
tion  through  this  surface. 

If  we  apply  Gauss's  Theorem  to  a  box  shut  in  by  a  tube 
of  force  and  the  portions  S^  S2  which  it  cuts  from  any  two 
equipotential  surfaces,  we  shall  have,  if  the  box  contains  no 
electricity, 

F.,X.,  —  F1S1=  0.  [163] 

where  Fl  and  F.2  are  the  average  values,  over  St  and  S2  respec 
tively,  of  the  normal  force  taken  in  the  same  direction  (that  in 
which  V  decreases)  in  both  cases.  In  other  words,  the  flow  of 
force  across  all  equipotential  sections  of  a  tube  of  force  con 
taining  no  electricity  is  the  same,  or  the  average  force  over  an 
oquipotential  section  of  an  empty  tube  of  force  is  inversely  pro 
portional  to  the  area  of  the  section. 


FIG.  41. 


When  a  tube  of  force  encounters  a  quantity  m  of  electricity 
(Fig.  41),  the  flow  of  force  through  the  tube  on  passing  this 


152 


ELECTROSTATICS. 


electricity  is  increased  by  ±irm.  If,  however,  the  tube  encoun 
ters  a  conductor  large  enough  to  close  its  end  completely,  a 
charge  m  will  be  found  on  the  conductor  just  sufficient  to  reduce 
to  zero  the  flow__of  force  (7)  through  the  tube.  That  is, 


47T 

It  is  sometinrelf  convenient  to  consider  an  electric  field  to  be 
divided  up  by  a  system  of  tubes  of  force,  so  chosen  that  the  now 
of  force  across  any  equipotential  surface  of  each  tube  shall  be 
equal  to  4?r.  Such  tubes  are  called  unit  tubes  ;*  for  wherever 
one  of  them  abuts  on  a  conductor,  there  is  always  the  unit  quan 
tity  of  electricity  on  that  portion  of  the  conductor's  surface  which 
the  tube  intercepts.  In  some  treatises  on  electricity  the  term 
"  line  of  force"  is  used  to  represent  a  unit  tube  of  force,  as 
when  a  conductor  is  said  to  cut  a  certain  number  of  "  lines  of 
force." 

It  is  evident  that  m  unit  tubes  abut  on  a  surface  just  outside 
a  conductor  charged  with  m  units*  of  either  kind  of  electricity, 
if  the  superficial  density  of  the  charge  has  everywhere  the  same 
sign.  These  tubes  must  be  regarded  as  beginning  at  the  con 
ductor  if  m  is  positive,  and  as  end&nfthpTe  if  m  is  negative. 
If  a  conductor  is  charged  at  some  places  with  positive  elec 
tricity  and  at  others  with  negative  electricity,  tubes  of  force 
will  begin  where  the  electrification  is  positive,  and  others  will 
end  where  the  electrification  is  nega 
tive. 

It  is  evident  that  no  tube  of  force 
can  return  into  itself. 


57.  Hollow  Conductors,  When  the 
nonconducting  cavity,  shut  in  by  a 
hollow  conductor  K  (Fig.  42),  contains 


FIG.  42. 


*  They  are  sometimes  called  "unit  Faraday  tubes,"  to  distinguish 
them  from  the  more  slender  tubes  of  unit  induction,  of  which  4  irm  start 
from  a  body  which  has  a  positive  charge  m. 


ELECTROSTATICS.  153 


quantities  of  electricity  (mlt  ra2,  m3)  etc.,  or  ^_^  m)  distributed 

in  any  way,  but  insulated  from  K,  there  is  induced  on  the 
walls  of  the  cavity  a  charge  of  electricity  algebraically  equal 
in  quantity,  but  opposite  in  sign,  to  the  algebraic  sum  of  the 
electricities  within  the  cavity. 

Call  the  outside  surface  of  the  conductor  S0  and  its  charge 
J/0,  the  boundary  of  the  cavity  St  and  its  charge  3/<,  and  sur 
round  the  cavity  by  a,  closed  surface  S,  every  point  of  which  lies 
within  the  substance  of  the  conductor,  where  the  resultant  force 
is  zero.  Now  the  surface  integral  of  normal  force  taken  over 
S  is  zero,  so  that,  according  to  Gauss's  Theorem,  the  algebraic 
sum  of  the  quantities  of  electricity  within  the  cavity  and  upon 
St  is  zero.  That  is, 

)  =0,        [164] 


and  this  is  our  theorem,  which  is  true  whatever  the  charge  on 
S0  is,  and  whatever  distribution  of  free  electricity  there  may 
be  outside  K.  If  the  distribution  of  the  electricity  within  the 
cavity  be  changed  by  moving  ??ij,  m2,  etc.,  to  different  positions, 
the  distribution  of  Jl/jon^,-  will  in  general  be  changed,  although 
its  value  remains  unchanged. 

If  A"  has  received  no  electricity  from  without,  its  total  charge 
must  be  zero  ;  that  is, 


If  a  charge  algebraically  equal  to  J/  be  given  to  /i, 
M^M-Mt. 

The  combined  effect  of  f  (??i),the  electricity  within  the  cavity, 

and  J/i,  the  electricity  on  the  walls  of  the  cavity,  is  at  all  points 
without  St  absolutely  null.  For,  if  we  apply  [153]  to/$,  any  sur 
face  drawn  in  the  conductor  so  as  to  enclose  S{,  we  shall  have  D,y 
everywhere  zero,  since  the  potential  function  is  constant  within 
the  conductor  ;  this  showrs  that  FI,  the  potential  function  due  to 


154  ELECTROSTATICS. 

all  the  electricity  within  $,  must  be  zero  at  all  points  without  S ; 
but  S  may  be  drawn  as  nearly  coincident  with  St  as  we  please. 
Hence  our  theorem,  which  shows  that,  so  far  as  the  value  of  the 
potential  function  in  the  substance  of  the  conductor  or  outside 
it,  and  so  far  as  the  arrangement  of  M0  and  of  M r,  any  free 

electricity  there  may  be  outside  /f,  are  concerned,  Mt  and  N    (m) 

might  be  removed  together  without  changing  anything.  The 
potential  function  at  all  points  outside  St  is  to  be  found  by  con 
sidering  only  M  and  M1. 

If  Si  happens  to  be  one  of  the  equipotential  surfaces  of  N    (m) 

considered  by  itself,  Mt  will  be  arranged  in  the  same  way  as  a 
charge  of  the  same  magnitude  would  arrange  itself  on  a  con 
ductor  whose  outside  surface  was  of  the  shape  $,',  if  removed 
from  the  action  of  all  other  free  electricity. 

The  potential  function  ( F2)  due  to  M0  and  M'  is  constant 
everywhere  within  S0 ;  for  if  we  apply  [154J  to  a  surface  £, 
drawn  within  the  substance  of  the  conductor  as  near  S0  as  we 
like,  we  shall  have 

FS-FO  =  O, 

which  proves  the  theorem. 

The  potential  function  within  the  cavity  is  equal  to  F2  +  Fj, 
where  Fi  is  the  potential  function  due  to  Mt  and  ^  (m).  Of  these, 

F2  is,  as  we  have  seen,  constant  throughout  K  and  the  cavity 
(Section  31)  which  it  encloses,  while  FI  has  different  values  in 
different  parts  of  the  cavity,  and  is  zero  within  the  substance  of 
the  conductor. 

Suppose  now  that,  by  means  of  an  electrical  machine,  some 
of  the  two  kinds  of  electricity  existing  combined  together  in  a 
conductor  within  the  cavity  be  separated,  and  equal  quantities 
(g)  of  each  kind  be  set  free  and  distributed  in  any  manner 
within  the  cavity. 

The  value  of  V}  within  the  cavity  will  probably  be  different 
from  what  it  was  before,  but  F2  will  be  unchanged  ;  for  the 


ELECTKOSTATICS.  155 

quantity  of  matter  in  the  cavity  is  unchanged,  being  now,  alge 
braically  considered. 


so  that  J/,  is  unchanged,  although  it  may  have  been  differently 
arranged  on  $,,  in  order  to  keep  the  value  of  Fi  zero  within 
the  substance  of  the  conductor.  If  now  a  part  of  the  free 
electricity  in  the  cavity  be  conveyed  to  St  in  some  way,  the  sub 
stance  of  the  conductor  will  still  remain  at  the  same  potential  as 
before.  For,  if  I  units  of  positive  electricity  and  n  units  of 
negative  electricity  be  thus  transferred  to  /S',-,  the  whole  quantity 

of  free  electricity  within  the  cavity  will  be  N    (m)  —  ?-(-»,  and 

that  on  Sf  will  be  3/4  -+-  /  —  n  :  but  these  are  numerically  equal, 
but  opposite  in  sign,  and  the  charge  on  $<,  if  properly  arranged, 
suffices,  without  drawing  on  3/0  to  reduce  to  zero  the  value  of 
FI  in  K.  Since  3/0  and  3/r  remain  as  before,  F2  is  unchanged, 
and  the  conductor  is  at  the  same  potential  as  before.  So  long 
as  no  electricity  is  introduced  into  the  cavity  from  without  K* 
no  electrical  changes  within  the  cavity  can  have  any  effect  out 
side  S^. 

Most  experiments  in  electricity  are  carried  on  in  rooms,  which 
we  can  regard  as  hollows  in  a  large  conductor,  the  earth.  F2, 
the  value  of  the  potential  function  in  the  earth  and  the  walls  of 
the  room,  is  not  changed  by  anything  that  goes  on  inside  the 
room,  where  the  potential  function  is  F=  F!  +  F2.  Since  we 
are  generally  concerned,  not  with  the  absolute  value  of  the  poten 
tial  function,  but  only  with  its  variations  within  the  room,  and 
since  F2  remains  always  constant,  it  is  often  convenient  to  dis 
regard  Fo  altogether,  and  to  call  FI  the  value  of  the  potential 
function  inside  the  room.  When  we  do  this  we  must  remember 
that  we  are  taking  the  value  of  the  potential  function  in  the 
earth  as  an  arbitrary  zero,  and  that  the  value  of  FI  at  a  point  in 
the  room  really  measures  only  the  difference  between  the  values 
of  the  potential  function  in  the  earth  and  at  the  point  in  ques 
tion.  When  a  conductor  A  in  the  room  is  connected  with  the 


156  ELECTROSTATICS. 

walls  of  the  room  by  a  wire,  the  value  of  Vj  in  A  is,  of  course, 
zero,  and  A  is  said  to  have  been  put  to  earth. 

58.  Induced  Charge  on  a  Conductor  which  is  put  to  Earth. 
Suppose  that  there  are  in  a  room  a  number  of  conductors,  viz.  : 
AI  charged  with  Ml  units  of  electricity,  and  Az,  A-^  A4,  etc., 
connected  with  the  walls  of  the  room,  and  therefore  at  the  po 
tential  of  the  earth,  which  we  will  take  for  our  zero.  If  the 
potential  function  has  the  value  pv  inside  A^  every  point  in  the 
room  outside  the  conductors  must  have  a  value  of  the  potential 
function  lying  between  ^  and  0,  else  the  potential  function  must 
have  a  maximum  or  a  minimum  in  empty  space.  If  pt  is  posi 
tive,  there  can  be  no  positive  electricity  on  the  other  conductors  ; 
for  if  there  were,  lines  of  force  must  start  from  these  conductors 
and  go  to  places  of  lower  potential ;  but  there  are  no  such  places, 
since  these  conductors  are  at  potential  zero,  and  all  other  points 
of  the  room  at  positive  potentials.  In  a  similar  way  we  may 
prove  that  if  pr  is  negative,  the  electricity  induced  on  the  other 
conductors  is  wholly  positive. 

Now  let  us  apply  [154B]  to  a  spherical  surface,  drawn  so  as 
to  include  A}  and  at  least  one  of  the  other  conductors,  but  with 
radius  a  so  small  that  some  parts  of  the  surface  shall  lie  within 
the  room.  If  we  take  the  point  0  at  the  centre  of  this  surface, 
we  shall  have 

=  !  CDrV-ds  +  -9  Cvds.  [165] 

aJ  o?J 

If  M  is  the  whole  quantity  of  electricity  within  the  spherical 
surface,  there  must  be  a  quantity  —M  outside  the  surface,  either 
on  the  walls  of  the  room  or  on  conductors  within  the  room. 
The  value  at  0  of  the  potential  function,  V<>,  due  to  the  elec 
tricity  without  the  sphere,  is  less  in  absolute  value  than , 

for  it  could  only  be  as  great  as  this  if  all  the  electricity  outside 
the  sphere  were  brought  up  to  its  surface. 
By  Gauss's  Theorem, 


ELECTROSTATICS.  157 

therefore,  Cvds  =  4  *«  \_M  +  a  TV] .  [166] 

Now,  if  Ml  is  positive,  the  integral  is  positive,  for  all  parts  of 
the  spherical  surface  within  the  room  yield  positive  differentials, 
and  all  other  parts  zero,  so  that  the  second  side  of  the  equation 
is  positive.  But  aF2  is  of  opposite  sign  to  J/,  and  is  less  in 
absolute  value  ;  hence,  J/  is  positive,  and  the  total  amount  of 
negative  electricity  induced  on  the  other  conductors  within  the 
spherical  surface  by  the  charge  on  A^  is  numerically  less  than 
this  charge,  unless  some  one  of  these  conductors  surrounds  A2 ; 
in  which  case  the  induced  chargo  comes  wholly  on  this  conduc 
tor,  while  the  other  conductors,  and  the  walls  of  the  room,  are 
free.  Some  of  the  tubes  of  force  which  begin  at  Al  end  on  the 
walls  of  the  room,  provided  these  latter  can  be  reached  from 
A1  without  passing  through  the  substance  of  any  conductor. 

59.  Coefficients  of  Induction  and  Capacity.  If  a  number  of 
insulated  conductors,  A2,  A~,  A4,  etc.,  are  in  a  room  in  the  pres 
ence  of  a  conductor  A^  charged  with  3/j  units  of  electricity,  the 
whole  charge  on  each  is  zero  ;  but  equal  amounts  of  positive  and 
negative  electricity  are  so  arranged  by  induction  on  each,  that 
the  potential  function  is  constant  throughout  the  substance  of 
every  one  of  the  conductors. 

Let  the  values  of  the  potential  functions  in  the  system  of  con 
ductors  be  2ht  Pzi  P^  Pn  etc.  Since  each  conductor  except  AI  is 
electrified,  if  at  all,  in  some  places  with  positive  electricity,  and 
in  others  with  negative  electricity,  some  lines  of  force  must 
start  from,  and  others  end  at,  every  such  electrified  conductor, 
so  that  there  must  be  points  in  the  air  about  each  conductor  at 
lower  and  at  higher  potentials  than  the  conductor  itself.  But 
the  value  of  the  potential  function  in  the  walls  of  the  room  is 
zero,  and  there  can  be  no  points  of  maximum  or  minimum  poten 
tial  in  empty  space  ;  so  that  />,  must  be  that  value  of  the  poten 
tial  function  in  the  room  most  widely  different  from  zero,  and 
Pn  Put  P^  etc.,  must  have  the  same  sign  as  />,. 

The  reader  may  show,  if  he  likes,  that  both  the  negative  part 


158  ELECTKOSTATICS. 

and  the  positive  part  of  the  zero  charge  of  any  conductor,  ex 
cept  AH  is  less  than  M±. 

Let  pn  be  the  value  of  the  potential  function  in  a  conductor 
AI  charged  with  a  single  unit  of  electricity,  and  standing  in 
the  presence  of  a  number  of  other  conductors  all  uncharged 
and  insulated.  Then  if  ^12,  _p18,  Pm  etc.,  are,  under  these  cir 
cumstances,  the  values  of  the  potential  functions  in  the  other 
conductors,  A<>,  A&  A^  etc.,  the  potential  functions  in  these 
conductors  will  be  Ml'p\^  M\P\zi  ^f\P\\->  etc.,  if  Al  be  charged 
with  Ml  units  of  electricity  instead  of  with  one  unit.  This  is 
evident,  for  we  may  superpose  a  number  of  distributions  which 
are  singly  in  equilibrium  upon  a  set  of  conductors,  and  get  a 
new  distribution  in  equilibrium  where  the  density  is  the  sum  of 
the  densities  of  the  component  distributions,  and  the  value  of 
the  resulting  potential  function  the  sum  of  the  values  of  their 
potential  functions. 

If  AI  be  discharged  and  insulated,  and  a  charge  M2  be  given 
to  A2,  the  values  of  the  potential  functions  in  the  different  con 
ductors  ma  be  written 


If  now  we  give  to  A1  and  A2  at  the  same  time  the  charges  M\ 
and  M2  respectively,  and  keep  the  other  conductors  insulated, 
the  result  will  be  equivalent  to  superposing  the  second  distribu 
tion,  which  we  have  just  considered,  upon  the  first,  and  the  con 
ductors  will  be  respectively  at  potentials, 

Mlpn-\-M.2p2l,     Mipu  +  Mzpzi,     Mtpu  +  ^>fe   etc.       [167] 

If  all  the  conductors  are  simultaneously  charged  with  quanti 
ties  J/i,  Jfefa,  MM  M±,  etc.,  of  electricity  respectively,  the  value 
of  the  potential  function  on  Ak  will  be 

V  = 


Writing  this  in  the  form  Vk  =  a.k  +  Mkpkk,  we  see  that  if  the 
charges  on  all  the  conductors  except  A  be  unchanged,  a,  will  be 
constant,  and  that  every  addition  of  —  units  of  electricity  to 


ft* 


KLK<Ti:<  'STATICS.  159 

the  charge  of  Ak  raises  the  value  of  the  potential  function  in 
it  by  unity.  If  we  solve  the  n  equations  like  [168]  for  the 
charges,  we  shall  get  n  equations  of  the  form 

Mk  =  F!  glt+  T;  g,,  4-  T>.  <]Ak  +  •-  +  F,</u  +  -  +  VK  qnk,  [169] 
where  the  q's  are  functions  of  the  p'&. 

If  all  the  conductors  except  Ak  are  connected  with  the  earth, 
Mk  =  Vk  q^,  and  ?u.  is  evidently  the  charge  which,  under  these 
circumstances,  must  be  given  to  Ak  in  order  to  raise  the  value 
of  the  potential  function  in  it  by  unity.  It  is  to  be  noticed  that 

qkk  and  are  in  general  different. 

Pm 

The  charge  which  must  be  given  to  a  conductor  when  all  the 
conductors  which  surround  it  are  in  communication  with  the 
earth,  in  order  to  raise  the  value  of  the  potential  function  with 
in  that  conductor  from  zero  to  unity,  shall  be  called  the 
capacity  of  the  conductor.  It  is  evident  that  the  capacity  of  a 
conductor  thus  defined  depends  upon  its  shape  and  upon  the 
shape  and  position  of  the  conductors  in  its  neighborhood. 

60.  Distribution  of  Electricity  on  a  Spherical  Conductor. 
Considerations  of  symmetry  show  that  if  a  charge  J/  be  given 
to  a  conducting  sphere  of  radius  r.  removed  from  the  influence 
of  all  electricity  except  its  own,  the  charge  will  arrange  itself 
uniformly  over  the  surface,  so  that  the  superficial  density  shall 

be  everywhere  <r  =  — — • 
4*1* 

The  value,  at  the  centre  of  the  sphere,  of  the  potential  function 

due  to  the  charge  3/on  the  surface  is  — ,  and,  since  the  potential 

r 

function  is  constant  inside  a  charged  conductor,  this  must  be 
the  value  of  the  potential  function  throughout  the  sphere.  If  M 

is  equal  to  r,  —  =  1  :  hence  the  capacity  of  a  spherical  conductor 
r 

removed  from  the  influence  of  all  electricity  except  its  own.  is 
numerically  equal  to  the  radius  of  its  surface. 


160  ELECTROSTATICS. 

61.  Distribution  of  a  Given  Charge  on  an  Ellipsoid.  It  is 
evident  from  the  discussion  of  honueoids  in  Chapter  J.  that  a 
charge  of  electricity  arranged  (on  a  conductor)  m  the  form  of 
a  shell,  bounded  by  ellipsoidal  surfaces  similar  to  each  other 
(and  to  the  surface  of  the  conductor),  and  similarly  placed, 
would  be  in  equilibrium  if  the  conductor  were  removed  from  the 
action  of  all  electricity  except  its  own.  We  may  use  this  prin 
ciple  to  help  us  to  find  the  distribution  of  a  given  charge  on  a 
conducting  ellipsoid. 

Let  us  consider  a  shell  of  homogeneous  matter  bounded  by 
two  similar,  similarly  placed,  and  concentric  ellipsoidal  surfaces, 
whose  semi-axes  shall  be  respectively  a,  6,  c,  and  (l+a)a, 
(1  -j-a)6,  (1  -f  a)  c.  If  any  line  be  drawn  from  the  centre  of 
the  shell  so  as  to  cut  both  surfaces,  the  tangent  planes  to  these 
two  surfaces  at  the  points  of  intersection  will  be  parallel,  and 
the  distance  between  the  planes  is  pa,  where  p  is  the  length 
of  the  perpendicular  let  fall  from  the  centre  upon  the  nearer  of 
the  planes. 

If  p  is  the  volume  density  of  the  matter  of  which  the  shell  is 
composed,  the  mass  of  the  shell  is  M  —  ^irabc  [(1-h  a)3  —  1]  p, 
and  the  rate  at  which  the  matter  is  spread  upon  the  unit  of  sur 
face  is,  at  any  point,  a  =  pS,  where  8  is  the  thickness  of  the 
shell  measured  on  the  line  of  force  which  passes  through  the 
point  in  question.  Eliminating  p  from  these  equations,  we  have 

M8  r 

2a 


If,  now,  in  accordance  with  the  hypothesis  that  the  thickness  of 
the  electric  charge  on  a  conductor  is  inappreciable,  we  make  a 
smaller  and  smaller,  noticing  that  8  differs  from  pa  by  an  infini 
tesimal  of  an  order  higher  than  the  first,  we  shall  have  for  a 
strictly  surface  distribution, 


If  the  equation  of  the  surface  of  the  ellipsoidal  conductor  is 


ELECTROSTATICS.  161 

we  have 


1  _  ,  , 

+ 


and 


* 


This  last  expression  shows  that,  as  c  is  made  smaller  and 
smaller,  o-  approaches  more  and  more  nearly  the  value 

M 

±rilb    /i-g-lT  [172] 

\         a-       lr 

and  this  gives  some  idea  of  the  distribution  on  a  thin  elliptical 
plate  whose  semi-axes  are  a  and  b. 

For  a  circular  plate,  we  may  put  a  =  b  in  the  last  expression, 
which  gives 

M 

[173] 


for  the  surface  density  at  a  point  r  units  distant  from  the  centre 
of  the  plate. 

The  charge  M  distributed  according  to  this  law  on  both  sides 
of  a  circular  plate  of  radius  a  raises  the  plate  to  potential 

so  that  the  capacity  of  the  plate  is 

— 

7T 

62.  Spherical  Condensers.  If  a  conducting  sphere  A  of  radius 
/•  (Fig.  43)  be  surrounded  by  a  concentric  spherical  conducting 
shell  B  of  radii  r,  and  r0  and  charged  with  m  units  of  electricity 
while  B  is  uncharged  and  insulated,  we  shall  have 

(1)  the  charge  m  uniformly  distributed  upon  S,  the  surface 
of  the  sphere ; 

(2)  an  induced  charge  —  m  (Section  57)  uniformly  distributed 
upon  St,  the  inner  surface  of  B ; 


162  ELECTROSTATICS. 

(3)    a  charge  +ra  (since  the  total  charge  of  B  is  zero)  uni 
formly  distributed  on  80,  the  outer  surface  of  B. 


FIG.  43. 


The  value  at  the  centre  of  the  sphere  of  the  potential  function 

due  to  all  these  distributions  is  VA  =  —  —  —  +—  ,  and  this  is 

r        rt        r0 

the  value  of  V  throughout  the  conducting  sphere.     The  value  of 

m 
the  potential  function  in  B  is  VB  =  —  • 

'  o 

If  now  a  charge  M  be  communicated  to  U,  this  will  add  itself 
to  the  charge  m  already  existing  on  S0,  and  the  charge  on  St  will 
be  undisturbed.  The  values  of  the  potential  functions  in  the 
conductors  are  now 


and    r  = 


If  now  B  be  connected  with  the  earth  so  as  to  make  VB  =  0, 
the  charges  on  S  and  St  will  be  undisturbed,  but  the  charge  on 


77? 


* 

S0  will  disappear.      VA  is  now  equal  to  —  —  —  . 

If  A  were  uncharged,  and  B  had  the  charge  M,  this  charge 
would  be  uniformly  distributed  upon  S0,  for,  since  the  whole 
charge  on  S  is  zero,  the  whole  charge  on  #t  must  be  zero  also. 
It  is  easy  to  see  that$  and  £{must  both  be  in  a  state  of  nature, 
for  if  not,  lines  of  force  must  start  from  S  and  end  at  S{,  and 
others  start  at  S{  and  end  at  /S,  which  is  absurd. 


ELECTROSTATICS. 


163 


If  A  were  put  to  earth  by  means  of  a  fine  insulated  wire 
passing  through  a  tiny  hole  in  .5,  and  if  B  were  insulated  and 
charged  with  M  units  of  electricity,  we  should  have  a  charge  x 
on  $,  a  charge  —  x  on  S/.  and  a  charge  3/-J-  x  on  S0.  To  find 

.v,  we  need  only  remember  that  F4  =  -  —  -+~-| •  =  0,  whence 

.r  may  be  obtained.  ?i      ?0      /0 

If  B  be  put  to  earth,  and  A  be  connected  by  means  of  the  fine 
wire  just  mentioned,  with  an  electrical  machine  which  keeps  its 
prime  conductor  constantly  at  potential  Fi,  A  will  receive  a  charge 
//  and  will  be  put  at  potential  FI.  To  find  y,  it  is  to  be  noticed 
that  there  is  a  charge  —  y  on  £,,  and  no  charge  on  £fl,  which  is 

y    y 

put  to  earth.      VA  = =  Fi,  whence  y  mav  be  obtained. 

r      r{ 

If  r=  99  millimeters  and  rt=  100  millimeters,  y  =  9900  Fx. 

If  a  sphere,  equal  in  size  to  A  but  having  no  shell  about  it, 
were  connected  with  the  same  prime  conductor,  it  too  would 
receive  a  charge  z  sufficient  to  raise  it  to  potential  FI,  and  z 
would  be  determined  by  the  equation  Vl=--  If  r  =  99,  we  have 
z  =  99  F!  ;  hence  we  see  that  A,  when  surrounded  by  B  at 
potential  zero,  is  able  to  take  one  hundred  times  as  great  a 
charge  from  a  given  machine  as  it  could  take  if  B  were  removed. 
In  other  words,  B  increases  ^4's  capacity  one  hundred  fold. 
A  and  B  together  constitute  what  is  called  a  condenser. 


FIG.  44. 


If  A  of  the  condenser  AB,  both  parts  of  which  are  supposed 
uncharged,  be  connected  by  a  fine  wire  (Fig.  44)  with  a  sphere 


164 


ELECTROSTATICS. 


A  which  has  the  same  radius  as  A,  and  is  charged  to  potential 
F!,  A  and  A'  will  now  be  at  the  same  potential  [F2],  and  A  will 
have  the  charge  a;,  and  ^4'  the  charge  ?/.     The  total  quantity  of 
electricity  on  A'  at  first  was  rFi,  so  that  a;  +  ?/  =  ?'Fi,  and 
_  ?/  _  a;      a;       a; 

i  o 

whence  a?  and  y  may  be  found. 

The  reader  may  study  for  himself  the  electrical  condition  of 
the  different  parts  of  two  equal  spherical  condensers  (Fig.  45) , 


FIG.  45. 

of  which  the  outer  surface  £!„  of  one  is  connected  with  an  elec 
tric  machine  at  potential  F15  and  the  inside  of  the  other,  $',  is 
connected  with  the  earth.  The  two  condensers,  which  are  sup 
posed  to  be  so  far  apart  as  to  be  removed  from  each  other's 
influence,  illustrate  the  case  of  two  Leyden  jars  arranged  in 
cascade. 

63.  Condensers  made  of  Two  Parallel  Conducting  Plates. 
Suppose  two  infinite  conducting  planes  A  and  B  to  be  parallel 
to  each  other  at  a  distance  a  apart ;  choose  a  point  of  the 
plane  A  for  origin,  and  take  the  axis  of  x  perpendicular  to  the 
planes,  so  that  their  equations  shall  be  a;  =  0  and  x  =  a.  Let  the 
planes  be  charged  and  kept  at  potentials  VA  and  VB  respectively. 
It  is  evident  from  considerations  of  symmetry  that  the  potential 
function  at  the  point  P  between  the  two  planes  depends  only 
upon  P's  x  coordinate,  so  that 

DF=0,    /),F=0,     A,2F=0,    ZVF=0. 


ELECTROSTATICS.  165 

Laplace's  Equation  gives,  then, 

D;-V=Q, 

whence  DXV=C,    and    V=Cx  +  D. 

If  x  =  0,  V=  VA  ;  and  if  x  =  rt,  V=  VB  ;  so  that 

F=F-F,      +  F     and  />,!" 


The  lines  of  force  are  parallel  between  the  planes,  and  the 
surface  densities  of  the  charges  on  A  and  B  are 

V  —  V  V  —  V 

—  -  and  —  --  -  respectively. 

4  TTCl  4  TTCt 

If  we  take  a  portion  of  area  S  out  of  the  middle  of  each  plate, 

there  will  be  a  quantity  of  electricity  on  *S'4  equal  to  —  *     A  ~    *', 

4:ira 

and  an  equal  quantity  of  the  other  kind  of  electricity  on  SB. 
The  force  of  attraction  between  SA  and  SB  will  be  2  ir<r  •  S.  or 

s  (VB-VAY 

STT  a- 

If  SB  be  put  to  earth,  the  charge  that  must  be  given  to  SA  in 
order  to  raise  it  to  potential  unity  is 

S    ' 


In  other  words,  the  capacity  of  SA  is  inversely  proportional  to 
the  distance  between  the  plates. 

In  the  case  of  two  thin  conducting  plates  placed  parallel  to  and 
opposite  each  other,  at  a  distance  small  compared  with  their 
areas,  the  lines  of  force  are  practically  parallel  except  in  the 
immediate  vicinity  of  the  edges  of  the  plates  ;*  and  we  may  infer 


*  See  Maxwell's  Treatise  on  Electricity  and  Maynetism,  Vol.  I.  Fig.  XII. 


166  ELECTROSTATICS. 

from  the  results  of  this  section  that  the  capacity  of  a  condenser 
consisting  of  two  parallel  conducting  plates  of  area  £,  separated 
by  a  layer  of  air  of  thickness  a,  when  one  of  its  plates  is  put  to 

S  V 

earth  is  very  approximately  -  for  large  values  of  -  • 

4?ra  a 

64.  Capacity  of  a  Long  Cylinder  surrounded  by  a  Concentric 
Cylindrical  Shell.  In  the  case  of  an  infinite,  conducting  cylinder 
of  radius  ?*(,  kept  at  potential  Vt  and  surrounded  by  a  concentric 
conducting  cylindrical  shell  of  radii  r0  and  rr,  kept  at  potential 
Vn,  we  have  symmetry  about  the  axis  of  the  cylinder,  so  that 
D^  V—  0,  and  Laplace's  Equation  reduces  to  the  form 


whence,  for  all  points  of  empty  space  between  the  cylinder  and 
its  she11' 


But   V=Vt  I  when  r  =  r,,  and  F=  F_  whenr  =  ?-0, 

p;iog£+F.u>g£ 

hence  F=  -  ^  -  =A  [175] 


and 


FIG.  47. 


The  surface  densities  of  the  electricity  on  the  outer  surface 
of  the  cylinder  and  the  inner  surface  of  the  shell  are  respectively 


ELECTROSTATICS.  167 

V*     and      '•-*. 


so  that  the  charge  on  the  unit  of  length  of  the  cylinder  is 

V  •  —  V 
—  1  -  —i  and  the  charge  on  the  corresponding  portion  of  the 


inner  surface  of  the  shell  is  the  negative  of  this.     We  may 
find  the  capacity  of  the  unit  length  of  the  cylinder  by  putting 

V0  =  0  and  Ft  =  1,  whence  capacity  =  -  —  • 


If  r0  in  this  expression  is  made  very  large,  the  capacity  of 
the  cylinder  will  be  very  small. 

In  the  case  of  a  fine  wire  connecting  two  conductors,  r{  will 
be  very  small,  and  there  will  be  no  conducting  shell  nearer 
than  the  walls  of  the  room,  so  that  the  capacity  of  such  a 
wire  is  plainly  negligible. 

65.  Charge  induced  on  a  Sphere  by  a  Charge  at  an  Outside 
Point,  The  value  at  any  point  P  of  the  potential  function  due 
to  iml  units  of  positive  electricity  concentrated  at  a  point 
AU  and  m2  units  of  negative  electricity  concentrated  at  a  point 
Az,  is 

V  =  —  -  —  2,    where    r,  =  A,P  and  rz  =  A,P. 
ri          rz 

It  is  easy  to  see  that  if  ml  is  greater  than  m^  so  that 


where  A  >  1,  V  will  be  equal  to  zero  all  over  a  certain  sphere 
which  surrounds  A2. 

If  (Fig.  48)  we  let  A,A2  =  a,   A10  =  8lf   AZ0  =  8,,    OD  =  r, 
it  is  easy  to  see  that 

X-a  a  0  g*Xg 

6l~  r":- 


168  ELECTROSTATICS. 


and  a  =       =      =  r.  [176] 

Oj  02 

If  PR  represents  the  force  ft  due  to  the  electricity  at  A^  and 
PQ  the  force  fa  due  to  the  electricity  at  A2,  the  line  of  action  of 
the  resultant  force  F  (represented  by  PL)  must  pass  through 
the  centre  of  the  sphere,  since  the  surface  of  the  sphere  is  equi- 
potential. 


FIG.  48. 

The  triangles  A1PO  and  A2PO  are  mutually  equiangular,  for 
they  have  a  common  angle  A±OP,  and  the  sides  including  that 
angle  are  proportional  (?*2  =  8i82)-  Hence,  from  the  triangles 
QPL  and  ALPA2,  by  the  Theorem  of  Sines, 

-  [177] 


sin  a2      sin  c^      sin  (a2  —  c^) 
whence  F==&==™h oArn^  p179-. 

*•  4*    £•''*•  3 

Now,  according  to  Section  49,  we  may  distribute  upon  the 
spherical  surface  just  considered  a  quantity  w2  of  negative  elec 
tricity  in  such  a  way  that  the  effect  of  this  distribution  at  all 
points  outside  the  sphere  shall  be  equal  to  the  effect  of  the 
charge  —  ra2  concentrated  at  A2<,  and  the  effect  at  points  within 
the  sphere  shall  be  equal  and  opposite  to  the  effect  of  the  charge 
raj  concentrated  at  At.  Since  F  is  the  force  at  P  in  the  direc- 


ELECTROSTATICS.  169 

tion  of  the  interior  normal  to  the  sphere,  we  shall  accomplish 
this  if  we  make  the  surface  density  at  every  point  equal  to  cr, 
where 

4TO.  =  -F=  -"*»'i  =  -(8i'-OM!;  [180] 

ra  rra 

and  if  we  now  take  away  the  charge  at  ./L,  the  value  of  the  po 
tential  function  throughout  the  space  enclosed  by  our  spherical 
surface,  and  upon  the  surface  itself,  will  be  zero.  If  the  spheri 
cal  surface  were  made  conducting,  and  were  connected  with  the 
earth  by  a  fine  wire,  there  would  be  no  change  in  the  charge  of 
the  sphere,  and  we  have  discovered  the  amount  and  the  distri 
bution  of  the  electricity  induced  upon  a  sphere  of  radius  r,  con 
nected  with  the  earth  by  a  fine  wire  and  exposed  to  the  action 
of  a  charge  of  MJ  units  of  positive  electricity  concentrated  at  a 
point  at  a  distance  ^  from  the  centre  of  the  sphere. 

If  now  we  break  the  connection  with  the  earth,  and  distribute 
a  charge  m  uniformly  over  the  sphere  in  addition  to  the  present 
distribution,  the  potential  function  will  be  constant  (although 
no  longer  zero)  within  the  sphere,  and  we  have  a  case  of  equi 
librium,  for  we  have  superposed  one  case  of  equilibrium  (where 
there  is  a  uniform  charge  on  the  sphere  and  none  at  A^  upon 
another.  The  whole  charge  on  the  sphere  is  now 


"  «1 

and  the  value  of  the  potential  function  within  it  and  upon  the 
surface, 

17—  ^  i   mi  —  m 

y  —          |-  — —  —       « 

r        6j        r 

If  the  conducting  sphere  were  at  the  beginning  insulated  and 
uncharged,  we  should  have  M=  0,  and  therefore 

._Viif!\    and  r=^i.         ^i] 

ria     )  *>i 

If  we  have  given  that  the  conducting  sphere,  under  the  influ 
ence  of  the  electricity  concentrated  at  A±  is  at  potential  FI,  we 


170  ELECTROSTATICS. 


know  that  its  total  charge  must  be  V\r  --  ~>  and  its  surface 


It  is  easy  to  see  that  the  sphere  and  its  charge  will  be 
attracted  toward  Al  with  the  force 


and  the  student  should  notice  that,  under  certain  circumstances, 
this  expression  will  be  negative  and  the  force  repulsive. 

If  m-L  =  m2,  the  surface  of  zero  potential  is  an  infinite  plane, 
and  our  equations  give  us  the  charge  induced  on  a  conducting 
plane  by  a  charge  at  a  point  outside  the  plane. 

The  method  of  this  section  enables  us  to  find  also  the 
capacity  of  a  condenser  composed  of  two  conducting  cylindrical 
surfaces,  parallel  to  each  other,  but  eccentric  ;  for  a  whole  set 
of  the  equipotential  surfaces  due  to  two  parallel,  infinite 
straight  lines,  charged  uniformly  with  equal  quantities  per 
unit  of  length  of  opposite  kinds  of  electricity,  are  eccentric 
cylindrical  surfaces  surrounding  one  of  the  lines,  A^  and  leav 
ing  the  other  line,  AU  outside.  We  may  therefore  choose  two 
of  these  surfaces,  distribute  the  charge  of  Al  on  the  outer  of 
these,  and  the  charge  of  A2  on  the  inner,  by  the  aid  of  the 
principles  laid  down  in  Section  49,  so  as  to  leave  the  values 
of  the  potential  function  on  these  surfaces  the  same  as  before. 
These  distributions  thus  found  will  remain  unchanged  if  the 
equipotential  surfaces  are  made  conducting. 

The  reader  who  wishes  to  study  this  method  more  at  length 
should  consult,  under  the  head  of  Electric  Images,  the  treatises 
of  Gumming,  Maxwell,  Mascart,  Tarleton,  and  Watson  and 
Burbury,  as  well  as  original  papers  on  the  subject  by  Murphy 
in  the  Philosophical  Magazine,  1833,  p.  350,  and  by  Sir 
W.  Thomson  in  the  Cambridge  and  Dublin  Mathematical 
Journal  for  1848. 


ELECTROSTATICS.  171 

66.  The  Energy  of  Charged  Conductors.  If  a  conductor  of 
capacity  C,  removed  from  the  action  of  all  electricity  except  its 
own,  be  charged  with  3/j  units  of  electricity,  so  that  it  is  at 

potential  Vl  —  ^,  the  amount  of  work  required  to  bring  up  to 
C 

the  conductor,  little  by  little,  from  the  walls  of  the  room,  the 
additional  charge  Am,  is  A  IF,  which  is  greater  than  Vl  •  A3f  or 

1L  .  A3/,  and  less  than  (  Vl  +  Aj,  V)  •  AJ/  or  3/1  +  AJ/.  AJ/. 

If  the  charge  be  increased  from  Ml  to  J/2  by  a  constant  flow, 
the  amount  of  work  required  is  evidently 


The  work  required  to  bring  up  the  charge  M  to  the  conductor 
at  first  uncharged  is  then 

M2      CV-      MV 

so-—  —T-  [18o] 

This  is  evidently  equal  to  the  potential  energy  of  the  charged 
conductor,  and  this  is  independent  of  the  method  by  which  the 
conductor  has  been  charged. 

If,  now,  we  have  a  series  of  conductors  A^  A&  A^  etc.,  in  the 
presence  of  each  other  at  potentials  I7",,  F2,  V3,  etc.,  and  having 
respectively  the  charges  M±,  M21  M&  etc.,  and  if  we  change  all 
the  charges  in  the  ratio  of  x  to  1,  we  shall  have  a  new  state  of 
equilibrium  in  which  the  charges  are  x  J/1?  .i-J/2,  xMs,  etc.  ;  and 
the  values  of  the  potential  functions  within  the  conductors  are 
xV^  xV2.  a-T>  etc.  The  work  (ATF)  required  to  increase  the 
charges  in  the  ratio  x  +  Aa;  instead  of  in  the  ratio  x  is  greater 
than 

Aa-)  (xV2)  +  (M2  Aa?)  (xVs)  +  etc., 


or  «  Aaj[  Jtfi  F!  -f  3/o  F2  -f  3/s  F;  -f  etc.]  , 

and  less  than 

(.>:  -h  AJ;)  Aar  [  Ml  V,  +  J/2  F2  +  J/8  F3  -h  etc.]  ; 


172  ELECTROSTATICS. 

hence  the  whole  amount  of  work  required  to  change  the  ratio 


x1  ,      x2  . 
irom  —  to  —  is 


W2  -  Wl  =    *  ~       IM1  F!  +  M,  F2  +  Ms  Vz  +  etc.].    [186] 

If  in  this  equation  we  put  xl  —  0  and  x2  =  1,  we  get  for  the 
work  required  to  charge  the  conductors  from  the  neutral  state 
to  potentials  V1}  F2,  F"3, 

V),      [187] 

a  particular  case  of  the  general  formula  stated  in  Section  27. 
The  work  required  to  make  any  combination  of  changes  of 
charge  on  any  system  of  fixed  conductors  is  evidently  equal 
to  the  difference  between  the  intrinsic  energies  of  the  system 
in  its  original  and  hnal  states.  If  Vk,  Vk'  represent  the  initial 
and  final  potentials  on  the  kth  conductor,  and  ek  and  ek  the 
original  and  final  charges, 

Vkek. 

Since  the  final  energy  is  independent  of  the  manner  in  which 
the  changes  are  produced,  we  may  suppose  that  the  changes 
take  place  gradually  and  at  the  same  relative  rate  for  all  the 
conductors,  so  that  at  any  instant  the  charge  of  each  conductor 
has  received  the  same  fraction  of  its  whole  increment  or 
decrement  that  every  other  conductor  has  received,  it  being 
understood  that  in  the  general  case  some  charges  will  be  in 
creased  and  others  decreased.  At  the  instant  when  the  change 
accomplished  is  to  the  whole  change  as  x  :  1,  the  charge  of  the 
/tth  conductor  is  ek  +  x(ek'  —  ek),  and  the  value  of  the  poten 
tial  function  in  this  conductor  is  Vk  +  x(VJ  —  Vk).  In  order 
to  increase  x  by  Ax,  the  charge  must  be  increased  by  the 
amount  Ax  (ek  —  ek),  and  to  bring  this  up  from  infinity  an 
amount  of  work  equal  approximately  to 


ELECTROSTATICS.  173 

must  be  done,  and  for  the  whole  system  the  corresponding 
work  is  y\e*f  -  ek)[  Vk  +  x  (  Vk'  -  F<.)]  \x.  To  find  the  work 

required  to  bring  about  the  whole  change,  this  expression 
must  be  integrated  with  respect  to  x  between  the  limits  0 
and  1.  This  process  yields 

Vt+  FiO  (et'  -  *,), 


and  by  comparing  this  with  the  result  stated  above  we  learn 
that  we  ma    also  write 


E'-E  =  ±(  Vk'  -  T,)  (e,'  +  et). 

We  learn  incidentally  that   /^'^  •  =  /^ct  -^V>  and  we  see 

that  if  all  but  two  (A^  A2)  of  any  system  of  conductors  are 
either  put  to  earth  or  are  insulated  and  without  charge, 

«i'  Jri  +  e2'  Vz  =  <?!  JY  +  ez  F2'. 

If  ei  =  1,  ez  =  0,  ei'  =  0,  ez'  =  1,  7^2  =  r/,  and  if  Tx  -  1, 
JT,,  =  0,  IV  =  0,  r2'  =  1,  ei  =  ez,  so  that  a  unit  charge  given 
to  AI,  while  Az  is  uncharged  and  insulated,  raises  A^  to  the 
same  potential  that  Al  would  have  if  it  were  uncharged  and 
insulated  while  A2  had  a  unit  charge  ;  and  the  same  quantity 
of  electricity  is  induced  on  A2  when  it  is  put  to  earth,  while 
Al  is  charged  to  potential  unity  as  would  be  induced  on  A±  if 
it  were  put  to  earth  and  A2  charged  to  potential  unity.  Using 
the  notation  of  Section  59,  this  shows  that  prk  =  pkr  and  that 
qrk  =  qkr  •  we  may  write,  therefore, 


E  = 

-  i  (Pit?  +  p&*  +  p*p*  +  •  •  •  +  ^..O 


174  ELECTROSTATICS. 

If  the  conductors  are  fixed  so  that  the  p's  and  ^'s  are  con 
stant,  we  may  learn  from  differentiating  this  last  equation  that, 
if  all  the  charges  but  ek  are  kept  constant,  UCkE  =  Vk,  and  if 
the  values  of  the  potential  function  in  all  but  one  of  the 
conductors  (the  &th)  are  unchanged  DV]E=  ek. 

If  the  system  changes  its  configuration,  the  p's  and  <?'s  are 
in  general  changed,  and  we  learn  that  if  the  charges  are  kept 
constant  during  the  change, 


A-  /• 

but  that  if  by  suitable  changes  in  the  charges  the  potentials 
are  unchanged, 


In  the  latter  case,  A  Vk,  or  \A(er_prjfc)  =  0,  so  that 


,or 


or  2  A'_^  +  2  A"^  4- 


If,  therefore,  <j>  is  any  coordinate,  which  defines  the  con 
figuration, 

limit   /A'^7\  li 

" 


mit 


A  system  of  conductors  with  constant  charges  when  left 
to  itself  tends  to  obey  the  urgings  of  the  reciprocal  forces 
between  its  parts,  and  therefore  to  diminish  its  intrinsic 
energy.  If,  in  this  case,  the  single  coordinate  <f>  is  free  to 
change  and  is  increased  by  A<£,  the  energy  after  the  change 
is  E  -f-  A'^,  where  A'J£  is  really  negative.  The  mechanical 
work  done  by  the  forces  is  —  D^E-  A<£.  If,  now,  <f>  had  been 
changed  as  before  by  the  same  small  increment,  A<£,  while  the 
potentials  were  kept  constant  by  bringing  up  to  each  con 
ductor  from  without  the  necessary  quantity  of  electricity,  the 
energy  after  the  change  would  have  been  E  -f  A"^,  where 


ELECTROSTATICS.  175 

\"E  is  really  positive.  The  energy  has  therefore  increased 
by  an  amount  practically  equal  to  the  former  loss.  Practi 
cally  the  same  amount  of  mechanical  work  has  been  done  as 
before,  and  enough  energy  has  been  introduced  from  without 
to  do  this  work  and  to  add  an  equivalent  amount  besides  this 
to  the  potential  energy  of  the  system.  The  contribution, 
therefore,  from  outside  sources  is  about  2  A"^.  These  state 
ments  applied  to  a  small  change  in  <f>  are  based  on  the  exact 
equation  D^'E =  —  D^'E,  proved  above. 

67.  If  a  series  of  conductors  Al}  A2.  A3,  etc.,  are  far  enough 
apart  not  to  be  exposed  to  inductive  action  from  one  another, 
and  have  capacities  Ci,  C2,  (73,  etc.,  and  charges  3/i?  3/2,  3/3,  etc., 
so  as  to  be  at  potentials  FI,  F2,  F3,  etc.,  where  3/1=C1F1, 
3/2  =  CoF.,  3/3=  (73F3,  etc.,  we  may  connect  them  together  by 
means  of  fine  wires  whose  capacities  we  may  neglect,  and  thus 
obtain  a  single  conductor  of  capacity 


The  charge  on  this  composite  conductor  is  evidently 
l£+J£+J£  +  ...= 


and  if  we  call  the  value  of  the  potential  function  within  it  F,  we 
shall  have 


whence  F=    *  **+  •  -     '"'  [188] 

a  formula  obtained,  it  is  to  be  noticed,  on  the  assumption  that 
the  conductors  do  not  influence  each  other. 

The  energy  of  the  separate  charged  conductors  before  being 
connected  together  was 


V  <-A        U2        Us 

[189] 


176  ELECTROSTATICS. 

and  the  energy  of  the  composite  conductor  is 
M*  +  Ms  +  •  .  •)  (6\  Fj  +  C2  V 


4-  £2  +  Cs  +.  .  • 

a*WJ 

'        L     J 


which  is  always  less  than  'E,  unless  the  separate  conductors 
were  all  at  the  same  potential  in  the  beginning. 

68.  Specific  Inductive  Capacity.  In  all  our  work  up  to 
this  time  we  have  supposed  conductors  to  be  separated  from 
each  other  by  electrically  indifferent  media,  which  simply 
prevent  the  passage  of  electricity  from  one  conductor  to 
another.  We  have  no  reason  to  believe,  however,  that  such 
media  exist  in  nature.  Experiment  shows,  for  instance,  that 
the  capacity  of  a  given  spherical  condenser  depends  essentially 
upon  the  kind  of  insulating  material  used  to  separate  the 
sphere  from  its  shell,  so  that  this  material,  without  conduct 
ing  electricity,  modifies  the  action  of  the  charges  on  the  con 
ductors.  Insulators,  when  considered  as  transmitting  electric 
action,  are  sometimes  called  dielectrics. 

Given  two  condensers  of  any  shape,  geometrically  alike 
in  all  respects,  with  plates  separated  in  the  one  case  by  a 
homogeneous  dielectric,  A,  and  in  the  other  case  by  another 
homogeneous  dielectric,  7>,  the  ratio  of  the  capacities  is 
found  to  be  the  'same  whatever  the  shape  or  dimensions  of 
the  condensers  when  these  same  two  dielectrics  are  used.  If 
this  ratio  is  unity,  the  dielectrics  are  said  to  have  the  same 
electrical  inductivity  or  the  same  specific,  inductive  capacity. 
If  the  ratio  of  the  capacities  of  the  first  and  second  con 
densers  is  n,  A  is  said  to  have  an  inductivity  n  times  as  great 
as  that  of  B.  The  electrical  inductivity  of  dry  air  at  the 
standard  pressure  and  temperature  being  chosen  as  a  standard, 
the  electrical  inductivities  of  all  other  known  substances  are 


ELECTROSTATICS.  177 

positive  quantities  which  in  the  case  of  any  one  specimen, 
though  somewhat  dependent  upon  conditions  of  temperature 
and  pressure,  may  be  considered  independent  of  the  electrical 
stress  to  which  the  substance  may  be  exposed.  The  letter  //. 
is  often  used  to  represent  the  inductivity  of  a  medium.  It  is 
generally  assumed,  for  the  sake  of  definiteness,  that  outside  all 
the  material  media  upon  which  we  can  experiment,  the  ether 
extends  indefinitely  in  all  directions  and  the  inductivity  of 
the  ether  is  assumed  to  be  sensibly  the  same  as  that  of  air 
under  standard  conditions.  We  cannot  expect  that  a  non- 
homogeneous  dielectric  will  have  the  same  inductivity  through 
out,  so  that  in  the  general  case  we  must  assume  that  /u.  is  a 
function  of  the  space  coordinates.  The  vector  formed  by 

multiplying  the  force  by  the  scalar  quantity  -p-  is  sometimes 

4  7T 

called  the  displacement.  The  force  is  occasionally  called 
the  electrical  intensity,  or  the  electromotive  intensity. 

We  may  best  sum  up  the  results  of  experiments  upon  the 
behavior  of  dielectrics  in  electric  fields  by  stating  some  gen 
eral  equations  which  may  be  used  in  solving  any  problem. 
We  shall  find  it  convenient  to  write  down  first,  for  the  sake 
of  comparison,  the  simplified  forms  of  these  equations  which 
we  have  shown  to  be  characteristic  of  the  electric  field  about 
any  distribution  of  electricity  when  air  is  the  only  dielectric. 

If  X,  Tj  Z  are  the  force  components  parallel  to  the  axes, 
and  if  V  is  the  potential  function,  so  that 

X  =  -DXV.    Y  =  -VyV,    Z  =  -DZV, 

we  know  that  when  /*  =  1, 

(1)  DXX  +  D,t  Y  +  DZZ  =  +  4  Trp, 

except  at  surfaces  where  p  is  discontinuous. 

(2)  The  surface  integral  of  the  normal  (outward)  com 
ponent  of  the  force  taken  over  any  closed  surface  is  equal 
to  4?r  times  the  amount  of  matter  (algebraically  reckoned) 
within  the  surface  ;  or 


178  ELECTROSTATICS. 

(3)  At  a  charged  surface  all  the  tangential  components  of 
the  force  are  continuous,  but  the  normal  components  are  dis 
continuous  in  the  manner  indicated  by  the  equation 

-ZVi  +  JVa  =  +  4  TTO-, 

where  JV^  and  Nz  represent  the  normal  force  components  taken 
away  from  the  surface  on  both  sides.  If  the  charged  surface 
is  not  equipotential,  the  lines  of  force  which  cross  it  are  in 
general  refracted  ;  for,  if  fa  is  the  angle  which  a  line  of  force 
in  reaching  the  surface  makes  with  its  normal,  fa  the  angle 
which  the  same  line  makes  with  the  normal  on  leaving  the 
surface  on  the  other  side,  and,  if  2\  and  1\  are  the  tangential 
components  of  the  force,  2\  —  —  N-^  tan  fa,  T2  —  N2  tan  fa, 
and  since  T±  =  T2,  Nz  tan  fa  +  Nj_  tan  fa  —  0,  or,  since  the 
normal  component  is  discontinuous, 

(4  TTO-  —  Ni)  tan  fa  -f-  j^  tan  fa  =  0. 

(4)  V  so   vanishes  at    infinity   that   rV  and  r2Dt.V  have 
finite  limits. 

If  we  now  introduce  a  new  vector  (called  the  induction) 
equal  to  the  product  of  the  scalar  point  function  ^  and  the 
force,  we  may  write  down  a  set  of  equations,  very  like  those 
which  we  have  just  enumerated  and  equivalent  to  them  when 
/*  =  1,  which  will  give  the  force  components  and  the  potential 
function  in  terms  of  the  charges  when  /x  is  different  from 
unity  and  (in  the  general  case)  determined  by  different 
analytic  functions  of  the  space  coordinates  in  different  por 
tions  of  space. 

In  general, 

(1)  Dx  (p.X}  +  Dy  0*  Y)  +  Dz  (pZ)  =  +  4  ^  [191] 

at  every  point  in  space,  except  at  surfaces  where  either  p  or  /x, 
is  discontinuous.  Since  in  all  cases  X=  —  DXV,  Y  =  —  Dy  F, 
Z  =  —  Dz  F,  this  equation  may  be  written 


Dx  (pDx  V)  +  D,  (pDy  V)  +  Dz  (v.D2  V)  =  -  4  *p.     [192] 


ELECTROSTATICS.  179 

In  a  dielectric  of  uniform  inductivity  it  becomes 


(2)  The  integral,  taken  over  any  closed  surface,  of  the  out 
ward  normal  component  of  the  induction  is  equal  to  4  IT  times 
the  amount  of  matter  within  the  surface  or 


C 


[193] 

(3)  If  the  surface  of  separation  between  two  different  dielec 
trics  which  are  in  contact  with  each  other  has  a  charge  of 
superficial  density,  o-,  all  the  force  components  tangent  to  the 
surface  are  continuous.  If  /^  and  /x2  are  the  inductivities  of 
the  two  media,  the  normal  component  of  the  induction  is  dis 
continuous  in  the  manner  indicated  by  the  equation 

/A^Y!  -f-  /x2.V2  =  +  4  TTO-, 

or  nt  Dni  V  +  n*  I\  V  =  -  4  TTO-.  [194] 

If  this  surface  has  no  charge,  o-  =  0,  and  the  normal  component 
of  the  induction  is  continuous,  though  the  normal  force  com 
ponent  is  discontinuous:  evidently,  the  law  of  refraction  of 
the  lines  of  force  is,  in  this  case,  tan  fa  :  ^  =  tan  fa  :  /x2. 
Whether  or  not  o-  is  zero,  J\\  tan  fa  -f  JNT2  tan  fa  =  0.  At  a 
charged  surface  where  the  dielectric  is  continuous, 


(4)  F  is  everywhere  continuous,  and  it  so  vanishes  at  infin 
ity  that  r  Fand  r2Dr  V  have  finite  limits.  The  first  derivatives 
of  V  are  everywhere  continuous,  except  at  charged  surfaces 
and  surfaces  where  the  inductivity  is  discontinuous  :  here  the 
tangential  derivatives  of  V  are  continuous  and  the  normal 
derivatives  have  the  properties  just  discussed.  The  lines  of 
force  and  the  lines  of  induction  are  coincident.  It  is  well 
to  notice  that  what  we  have  here  called  the  induction  and 
what  is  usually  called  induction  in  perfectly  hard  magnets 
are  different  special  cases,  as  will  be  shown  later  on,  of  a 


180  ELECTROSTATICS. 

much  more  complex  vector  which  appears  in  some  general 
problems. 

It  is  easy  to  prove  with  the  help  of  [149]  a  series  of  theo 
rems  concerning  the  potential  function  analogous  to  those 
already  found  for  the  case  where  /t*  =  1.  For  instance  :  if  the 
closed  surface  Sl  shuts  in  the  closed  surface  Sz,  there  cannot 
be  two  different  functions,  V  and  V,  which  (1)  between  S1  and 
$2  satisfy  the  equation 

Dx  (pDxw)  +  Dy  (pDvw)  +  Dz  (f,Dxw)  =  0, 

where  /A  is  a  given,  everywhere  positive,  analytic  function  of 
the  space  coordinates,  (2)  are  continuous  in  that  region  with 
their  first  derivatives,  and  (3)  are  equal  at  every  point  of  Si 
and  Sz.  Assuming,  for  the  sake  of  argument,  that  two  such 
functions  exist,  we  may  call  their  difference  u  and  note  that 
u  and  its  first  derivatives  are  continuous  between  Si  and  SZJ 
and  that  u  vanishes  at  every  point  of  these  surfaces.  Since 
u  satisfies  the  equation 

+  D       Du  +  D  (*.Du  =  0 


between  Si  and  $2,  we  may  conveniently  make  A  =  p,  U=V=u 
in  [149],  for  both  integrals  in  the  second  member  of  the  equa 
tion  vanish,  and  we  learn  that 


when  extended  over  the  region  in  question.     Since  /t  is  posi 
tive,  and  the  integrand  can  never  be  negative, 

Dxu  =  Dyu  =  Dzu  =  0, 

and  u  is  a  constant.     But  u  =  0  on  Si  and  S2)  hence  V  and  V 
are  identical. 

If,  while  satisfying  conditions  (1)  and  (2),  V  and  V  are 
required  to  have  equal  normal  derivatives  at  every  point  of  Sl 
and  $2>  it  is  easy  to  prove  in  a  similar  manner  that  one  can 
differ  from  the  other  only  by  a  constant. 


ELECTROSTATICS.  181 

With  given  values  of  the  volume  density  in  given  regions 
of  space,  and  with  given  values  of  the  superficial  density  on 
given  surfaces,  the  force  components  and  the  potential  func 
tion  are,  in  general,  different  when  /x  =  1  and  when  /*,  is  dif 
ferent  from  1,  and,  if  the  dielectric  is  heterogeneous  with 
surfaces  of  discontinuity  in  tt,  not  equipotential  surfaces,  the 
forms  of  the  lines  of  force  are  very  different  in  the  two  cases. 

If  the  dielectric  of  a  given  condenser,  the  plates  of  which 
are  the  surfaces  Si  and  S2,  is  air,  and  if  these  plates  have  given 
charges,  V  must  satisfy  Laplace's  Equation  between  Si  and  £.„ 
while  at  every  part  of  the  condenser  plate  Dn  V  =  —  4  TTO-.  If, 
now,  a  homogeneous  dielectric  of  inductivity  /u.  be  substituted 
for  the  air,  the  new  potential  function  V  satisfies  Laplace's 
Equation  between  Si  and  Sa  (since  /x  is  constant  andp  is  zero), 

and  at  every  point  of  Si  or  S2,  DnV  =      —  •     Now   V / p. 

satisfies  all  these  last  conditions,  and  since  two  functions 
which  do  so  can  at  most  differ  by  a  constant,  we  may  write 

V  =  V/IL  +  C. 

The  force  in  any  direction  at  any  point  in  the  dielectric  is 
1  /tt  as  great  in  the  second  case  as  in  the  first.  If  /Si  and  S2, 
instead  of  having  given  charges,  had  been  kept  at  the  given 
potentials  f\  and  F2,  the  density  of  the  charge  at  any  point 
of  either  plate  would  have  been  tt  times  as  great  in  the  second 
case  as  in  the  first,  while  the  potential  function  (and  the 
force)  would  have  had  the  same  value  at  every  point,  which 
ever  dielectric  was  used.  The  capacity  of  the  condenser  is, 
in  this  case,  equal  to 


The  generalized  form  [192]  of  Poisson's  Equation,  when 
expressed  in  terms  of  the  orthogonal  curvilinear  coordinates 
u,  v,  w  as  independent  variables,  becomes 


182  ELECTROSTATICS. 

/*  [  V  •  A,2  F  +  V  •  D?  V  4-  A,,,2  •  D*  V+DUV- 


+  (/>„  F  •  I>xw  +  Dt,  V  •  Dxv  +  D,,  V  •  Dxw) 

(I)./*  •  />XN  +  />„/*  '  Dxv  +  A,/*  -  Dxw) 

+  (Du  F.  Dy«  +  Dv  V-  Dvv  +  Dw  F-  £>//;) 

(DM/u,  •  JD^w  +  />„/*•  J9yt;  +  Dw/x  •  D^w) 

+  (Du  F-  Z>2%  +  Dv  F-  Z>zv  4-  DM.,  F-  Dzw) 

(Z)Mfi  •  Dsu  +  DvtL  •  Dzv  +  Dwfi'  Dzw)  =  -±-rrp. 

If  /x  is  a  function  of  one  of  the  coordinates,  u,  only,  the 
family  of  surfaces  on  which  u  is  constant  are  possible  equi- 
potential  surfaces  due  to  a  distribution  of  electricity  in  this 
dielectric,  provided  the  special  form  of  the  equation  just 
stated,  obtained  by  putting 

DvfJL  =  DwfJL  =  D,  F  =  Dw  V  =  P  =  0, 
that  is,  provided  the  equation 


involves  only  w.  Now  DM/x//u,  is,  by  hypothesis,  a  function 
of  w  only,  so  that  the  condition  is  that  the  ratio  of  V2u  to  hu* 
shall  be  independent  of  v  and  w,  and  this  is  the  condition 
(Section  35)  that  must  be  satisfied  when  the  dielectric  is  air, 
in  order  that  the  surfaces  upon  which  u  is  constant  may  be 
possible  equipotential  surfaces. 

It  is  easy  to  see  that  if  the  space  between  two  equipoten 
tial  surfaces  in  air  about  a  distribution  of  electricity  be  filled 
with  a  dielectric  the  inductivity  of  which  is  either  constant 
or  else  a  function  only  of  the  parameter  of  the  original  equi 
potential  surfaces,  the  new  equipotential  surfaces  will  coincide 
with  the  old  ones,  though  the  value  of  the  potential  function 
on  any  particular  surface  will  generally  be  changed. 

If  in  [149]  we  make  u  =  F,  the  potential  function  due  to  any 


ELECTROSTATICS.  183 

distribution  of  electricity,  and  if  we  make  A.  =  /A,  we  may  apply 
the  equation  to  all  space  after  we  have  enclosed  by  pairs  of  new 
surfaces  all  surfaces  of  discontinuity  of  /a,  p,  or  DnV,  and 
learn  that  the  intrinsic  energy  of  the  distribution  is  equal  to 


extended  over  all  space. 


When  the  potential  function,  F,  due  to  a  given  distribution 
(p,  <r)  of  electricity  with  any  given  set  of  dielectrics  has  been 
found,  we  may  ask  what  distribution  (p',  a-')  of  electricity 
would  have  given  this  same  potential  function  if  all  the 
dielectrics  had  been  displaced  by  homogeneous  air.  The  dis 
tribution  (pf,  a-')  is  called  the  apparent  charge  to  distinguish 
it  from  the  distribution  (p,  a-)  which  is  sometimes  called  the 
real  or  the  intrinsic  charge.  From  the  apparent  charge  when 
found,  V  might  be  calculated  by  means  of  the  familiar  integrals 


When  V  is  given,  the  quantity  p'  is  determined  at  all  points 
where  the  equation  has  a  definite  meaning  by  V2F=  —  47rp' 
and  the  quantity  a-'  at  all  surfaces  where  the  normal  derivative 
of  V  is  discontinuous  by  the  equation  N±  +  -ZV2  =  4  TTO-'. 

Now     Dx^Dx^  +  Dy(t,DyV)  +  Dz^Dzr)=-^p, 
or     jtV2  V  +  (DXV-  Dxn+Dy  V-  Dy^+Dz  V-  Da/*)  = 
or  -47r/zp'+(.DxF.  DxtL+Dv  V-  DytL+Dz  V-  D^)  = 
and  this  defines  p'.    In  every  region  where  /u.  is  constant  p'  =  p//x. 

In  the  most  general  case  of  a  surface  where  the  normal 
derivative  of  V  is  discontinuous,  there  is  a  discontinuity  in  /x 
at  the  surface  and  a  charge,  <r,  on  the  surface  so  that 

^\   +  frNt  =  4  7TO-,     A\  +  N2  =  4  7TO-', 

whence     «<  =  -  +  ^  ~  ^  =  ^  +  ^2(/Al  ~  ^>.        [197] 


184  ELECTROSTATICS. 

In  a  particular  instance  there  may  be  a  surface  charge  with 
no  discontinuity  in  the  dielectric,  in  which  case  a-'  =  o-///, ;  or 
there  may  be  discontinuity  in  the  dielectric  with  no  real 
surface  charge,  in  which  case 

The  difference  (p'  —  /a,  o-'  —  o-)  between  the  apparent  charge 
and  the  intrinsic  charge  is  sometimes  called  the  induced 
charge. 

The  solving  of  one  or  two  simple  problems  will  suffice  to 
illustrate  the  use  of  the  general  equations  which  determine 
the  potential  function  when  the  dielectric  is  not  homogeneous. 

I.  "A  condenser  consists  of  two  concentric  conducting 
spherical  surfaces  of  radii  a  and  b  separated  by  a  dielectric 
the  inductivity  of  which  at  a  distance,  r,  from  the  common 

c  ~\~  r 
centre,  0,  of  the  spherical  surfaces  is The  inner  plate, 

of  radius  a,  has  a  charge  E.  The  outer  plate  is  at  potential 
zero.  The  potential  function  in  the  dielectric  is  evidently  a 
function  of  r  only ;  what  is  its  value  ?  " 

Since  the  induction  through  any  closed  surface  is  equal  to 
4?r  times  the  intrinsic  charge  within,  we  may  imagine  a 
spherical  surface  drawn  in  the  dielectric  with  centre  at  0  and 
radius  equal  to  r  and  then  assert  that,  if  F  is  the  force, 

c  4-  r  v 

4  Ti-r2  •  —Z—  -  F  =  4  TrE  so  that  F  =  —  Z>  V  = 


r  (c  -+-  ?•) 

and  V  —  —  log  — •      The  capacity  of  the  condenser  is 

c         r  (b  +  c) 

c  -f-  log  ~-7 £  •     The  apparent  surface  density  on  the  inner 

a (b  +  c) 

plate  is  a-'  =  E/\_kira(a  +  c)],  the  intrinsic  surface  density  is 
JZT/4?ra2,  and  the  density  of  the  charge  induced  at  the  inner 
surface  of  the  dielectric  is  —  EC  /  [±  IT  a*  (a  +  c)]. 

II.    "A  condenser  consists  of  two  large,  plane,  conducting 
plates  parallel  to  each  other  and  separated  by  three  slabs,  slt 


ELECTROSTATICS.  185 

52,  s3,  of  dielectric  of  thickness  a,  b,  and  c  respectively,  and  of 
inductivity  1,  /x,  and  1.  What  is  the  capacity  of  the  con 
denser  per  unit  area  of  one  of  its  plates  ?  " 

Take  the  axis  of  x  perpendicular  to  the  faces  of  the  plates 
with  the  origin  in  the  first  plate,  which  shall  be  kept  at 
potential  zero.  It  is  evident  that  the  potential  function  is  a 
function  of  x  only,  so  that  Dx2  V  =  0  in  each  slab  of  dielectric 
and  F  must  be  of  the  form  Lx  +  J/".  Denote  the  functions 
which  give  the  potential  in  the  three  slabs  by 


When  j-  =  0,  J\  =  0.  When  x  =  a,  -  DJ\  +  ^DZVZ  =  0. 
and  1\  =  F2.  When  x  =  (a  +  b),  -  pD,  F2  +  Dz  F3  =  0,  and 
F2  =  rs.  We  have,  therefore,  J\  =  L^, 

F2  =  A  (a:  +  a/t  -  a)  //i,  F,  =  A(/xx  +  &  -  fyt)//4- 
When  a:  =  0,  D^  V  =  —  4  TTO-  =  Llt  and,  if  F8  =  1  when 


x  =  a  +b  +  c,  we  get  <r  = 

47r(ft«  +  [J.C  +  b) 

and  this  is  the  capacity  per  unit  area  of  the  first  plate. 

69.  Polarized  Distributions.  Imagine  two  homogeneous 
bodies,  P  and  ^V,  of  equal  but  opposite  densities,  p  and  —  p, 
of  the  same  dimensions,  and  occu 
pying  at  the  same  time  the  same 
space,  in  which,  of  course,  the 
resultant  density  is  zero.  If  P  be 
moved  without  rotation  through 
a  small  distance  h,  in  some  direc 
tion,  there  will  be  a  space  of  no 
density  common  to  P  and  X,  a 
space  of  density  p  where  P  extends  FIG.  49. 

beyond  ^V,  and  a  space  of  density 

—  p  where  A"  extends  behind  P.  The  thickness  of  the  shell  of 
matter,  measured  on  the  exterior  normal  to  the  space  of  no 
density,  is  A«.  If,  now,  h  be  made  to  approach  zero,  and  p 


186  ELECTROSTATICS. 

be  imagined  to  increase  without  limit  so  as  to  keep  the  product 
pk  always  equal  to  a  given  constant  /, 


and  we  have  in  the  limit  merely  a  superficial  distribution,  of 
density  o-  =  /  cos  (A,  n),  on  the  boundary  of  the  space  originally 
occupied  in  common  by  P  and  N.  Since  the  direction  of  h  is 
fixed  in  space,  and  n  is  an  exterior  normal,  the  distribution 
consists  partly  of  negative  matter  and  partly  of  positive  matter 
in  equal  amounts.  The  surface  density  is  equal  to  zero  at 
points  of  contact  of  the  distribution  with  tangents  parallel  to 
the  direction  of  h. 

If  this  distribution  be  divided  up  into  filaments  parallel  to 
h,  it  is  clear  that  the  charges  on  the  ends  of  every  filament 
are  equal  and  opposite,  and  that  each  is  equal  in  amount  to 
ql,  where  q  is  the  cross-section  of  the  filament  in  question. 
It  is  easy  to  see  from  this  that  if  the  distribution  were 
placed  in  a  uniform  field  of  force  of  intensity  F,  this  field 
would  exert  upon  any  such  filament  of  length  I  a  couple  of 
moment  J^-sin(A,  F)  •  qll,  and  upon  the  whole  distribution  a 
couple  of  moment  .F-sin(7i,  F)  •  I  times  the  volume  of  the 
space  enclosed  by  the  distribution.  /  is,  therefore,  numeri 
cally  equal  to  the  moment  of  the  couple,  per  unit  of  volume, 
per  unit  field  perpendicular  to  the  direction  of  h.  The  dis 
tribution  just  described  is  said  to  be  a  uniformly  polarized 
distribution.  /  is  called  the  intensity  of  the  polarization. 

If,  for  instance,  P  is  a  sphere  of  radius  a  with  centre  at  0, 
and  if  r*  =  x2  4-  y'2  +  #2,  the  potential  function,  V(x,  y,  2), 
due  to  its  own  mass,  has,  as  we  know,  the  value  2  TT/O  (a2  —  1  r2) 
at  inside  points,  and  the  value  4:7rpa3/3r  at  outside  points. 
After  P  has  been  displaced  through  a  distance  h  parallel 
to  the  x  axis,  the  potential  function  at  any  point  (x,  y,  «), 
either  in  the  space  common  to  P  and  N  or  outside  both, 
has  the  value  V  (x  —  h,  ?/,  z)  —  V  (x,  y,  «).  If  the  point 
is  within  both  P  and  N,  the  value  of  this  quantity  is 
2  irph  (2  x  —  k)  1  3,  but  if  the  point  is  without  both  P  and 


ELECTROSTATICS. 


187 


X,  the  value  is  2  7rasph(2x  -  h)/3  ,*.  The  limits  of  these 
expressions  (4?r/.r/3  and  4=Tra3Ix/3  r3)  give  the  values  of  the 
potential  function  within  and  without  a  sphere  uniformly 
polarized  to  intensity  I  parallel  to  the  x  axis.  Within  the 
sphere  the  equipotential  surfaces  are  planes  perpendicular  to 
the  x  axis,  the  field  is  uniform,  and  since  .Y=  —  DXV,  the 
lines  of  force  are  parallel 
to  the  negative  direction 
of  the  x  axis.  Consider 
ations  of  symmetry  show 
that  the  lines  of  force 
without  the  sphere  are 
curves  lying  in  planes 
through  the  axis  of  x. 
From  the  expression  for 
V  at  outside  points  we 
learn  that  if  0  is  the 
angle  which  the  radius 
vector  drawn  from  the 
origin  to  any  point 
makes  with  the  x  axis, 
the  equipotential  sur 
faces  of  revolution  with 
out  the  sphere  may  be 
considered  as  generated 
by  plane  curves  which 
belong  to  the  family 
cos  6/i*  =  c.  Curves  of 
this  family  lying  in  a  FIG.  50. 

plane  are  cut  orthogo 
nally  by  curves  in  the  same  plane  which  have  the  equa 
tion  r=  &-sin'20,  and  this  evidently  gives  the  lines  of  force. 
Fig.  50  shows  the  forms  of  these  lines  and  the  direction  of  the 
force.  It  is  to  be  noted  that  this  direction  changes  abruptly 
at  the  surface ;  on  the  jc  axis  without  the  sphere  the  force  is 
directed  from  left  to  right,  whereas  within  the  sphere  it  is 


188  ELECTROSTATICS. 

directed  from  right  to  left.  This  discontinuity  is  wholly 
explained,  as  a  little  simple  computation  will  show,  by  the 
fact  that  at  any  superficial  distribution  of  density  o-  every 
tangential  component  of  the  force  is  continuous,  but  the 
normal  component  is  discontinuous  by  4  TTO-. 

The  potential  function  belonging  to  a  uniform  field  of  force 
of  intensity  X0,  the  lines  of  which  are  parallel  to  the  x  axis,  is 
—  XQx,  and  if  into  such  a  field  a  sphere  of  radius  a,  uniformly 
polarized  to  intensity  /  parallel  to  the  x  axis,  is  brought,  and 
if  we  define  the  constant  x  by  the  equation  XQ  =.  4  7r/x/3j  the 


FIG.  51. 

potential  function,  referred  to  the  centre  of  the  sphere  as 
origin,  will  have  the  value  47r/x(l  —  \)/3  at  points  within 
the  sphere,  and  the  value  4  TT/Z  [>3  /  3  (z2  +  y2  +  z2)3/2-  x/3] 
at  outside  points.  The  field  within  the  sphere  is  now  a  uni 
form  field  of  intensity  4  TT/(X  —  1)  /3  directed  parallel  to  the 
x  axis  :  if  x  =  1?  this  force  vanishes.  The  equipotential  sur 
faces  of  revolution  without  the  sphere  could  be  generated  by 
the  revolution  about  the  x  axis  of  a  family  of  curves  the 
equation  of  which  in  the  xy  plane  is  4  -rrlx  [«-s/3  r3  —  x/3]  —  c> 
where  r2  =  x2  -\-  if.  The  equation  of  the  family  of  curves 
which  cut  these  orthogonally  may  be  written, 

27r/?/2(2  a3/3  >'3  +  x/3)  =  m, 


ELECTROSTATICS.  189 

and  this  represents  the  lines  of  force.  These  lines  may  be 
easily  plotted  for  any  value  of  ^,  by  assuming  in  succession  a 
series  of  values  of  r  and  computing  the  corresponding  values  of 
?/.  Figs.  51  and  52  show  two  characteristic  forms  which 
the  lines  may  have.  In  the  first  ^  =  +  1,  in  the  second 
X  =  —  3.  Some  slight  theoretical  interest  attaches  to  the 
case  for  which  ^  =  —  2,  and  the  reader  may  care  to  plot 
for  himself  the  corresponding  curves.  He  should  indicate 
the  direction  of  the  force  at  various  points  by  arrows. 

The  value,  at  inside  points,  of  the  potential  function  due 
to  a  homogeneous  ellipsoid  of  density  p,  with  axes  coincident 


FIG.  52. 

with  the  coordinate  axes,  is  given  on  page  121.  If  we  call 
this  p  •  O  (z,  y,  z),  we  may  write 

O  (x,  y,  z)  =  abdr(G<>  -  A>2  -  L^f  -  H&*), 

where  G0,  K0,  L0,  MQ  have  the  same  values  at  all  points  of  the 
mass.  If,  now,  we  consider  an  ellipsoidal  distribution,  uni 
formly  polarized  to  intensity  /,  in  a  direction  s,  it  is  easy  to  see 
that  the  value  of  the  potential  function  within  the  distribution  is 

—  I[DlQ,  •  cos  (x,  s)  +Dy£l  -  cos  (y,  s)  +  Dzto  •  cos  (z,  s)] 
or  2  abcirI\K^  •  cos  (z,  s)  +  LQy  •  cos  (y,  s)  -f  M<p  •  cos  (z,  s)], 

and  that,  if  we  regard  the  polarization  as  a  vector  and  denote 
its  components  by  A,  B,  and  (7,  the  force  components  are 

-  2  TrabcAKfr  -  2  TrabcBLQ,  -  2  irabcCMQ. 

The  field  within  the  distribution  is,  therefore,  uniform,  and 
it  has  a  direction  defined  by  cosines  which  are  to  each  other 
as  A"0-cos  (x,  s)  :  L0-cos  (y,  s)  :  Jf0-cos  (z,  s).  It  is  to  be 
noticed  that  this  direction  does  not  coincide  with  that  of  the 


190  ELECTROSTATICS. 

polarization   unless  K0  =  L0  =  M0  (so  that  the    ellipsoid  is 
really  a  sphere),  or  unless  the  direction  of  polarization  coin 
cides  with  that  of  one  of  the  principal  axes  of  the  ellipsoid. 
In  certain  cases  the  elliptic  integral 


=  f 

*/0 


ds 


(s  +  a2)3/2(s  4-  62)1/2(s  +  c2)1/2 

and  the  corresponding  integrals  L0  and  M0  can  be  easily 
evaluated.  If,  for  instance,  a  =  b  =  c,  these  quantities  evi 
dently  have  the  common  value  2/3  a3. 

If  the  ellipsoid  is  a  figure  of  revolution,  we  may  find  the 
values  of  K0,  L0,  MQ  with  the  help  of  the  integrals 

/ds 
(s-M^O'  +  m2)1/2 

1  t  /(.s  +  m2)1/2  -  (m2  -  Z2)1 

~  (m:2  -  I2)1/'2'  °?\(j  +  m2)1/2  +  (m2  -  P)1 

+  m21/2 


/ 


^ 

-1 


1    r 
*  J  ( 

/o?s 
(s  +  ^2)(.9  +  m2)3/2 


1  (fa 


f  ds  \ 

J  (s  +  Z2)  (s  +  m2)1/2/ 


-  +  m2)1/2 
In  the   case  of  a  prolate    ellipsoid  where   «  >  b,  b  =  c,  and 


ELECTROSTATICS, 
and  the  force  components  within  the  ellipsoid  are 


1  — 


191 


( 
-, 


2,c(*  •'-*' 


l-A 
TT~e}' 


l- 


If,  while  &  is  constant,  a  be  increased  without  limit,  e 
approaches  the  limit  unity,  (1  —  e2)  -log[(l  +  e)/(l  —  e)] 
the  limit  zero,  and  the  ellipsoid  becomes  an  infinitely  long 
cylinder  of  revolution,  for  which  the  force  components  are 
0,  -2-rrB,  -2-rrC. 

In  the  case  of  an  oblate  ellipsoid  where  a  <  b,  b  =  c,  and 
e  =  ^/b-  -a'2/b, 


and  the  force  components  within  the  ellipsoid  are 


-4,^    -.- 


Vl  -  e2 


-2wC 


If,  while  b  and  e  are  constant,  a  is  made  to  approach  zero,  e 
approaches  the  limit  unity,  the  limiting  values  of  the  force 
components  are  —  4  -rrA,  0,  0,  and  we  have  the  case  of  a  circu 
lar  disc,  in  which,  if  the  direction  of  polarization  lies  in  the 
plane  of  the  disc,  the  resultant  force  is  zero. 


192  ELECTROSTATICS. 

If  the  imaginary  body  P,  instead  of  being  of  the  same 
density  throughout,  had  consisted  of  two  homogeneous  por 
tions  of  densities  p±  and  p2,  to  the  left  and  to  the  right  of 
their  surface  of  separation,  $;  if  the  density  of  N  had  been 
at  every  point  equal  and  opposite  to  that  of  P,  and  if  the 
limits  of  pji  and  pji  had  been  the  constants  ^  and  /2,  the 
resulting  surface  distribution,  on  the  boundary  of  the  space 
occupied  originally  by  N  and  P  conjointly,  would  have  had 
the  density  o-  =  It  cos  (h,  n)  to  the  left  of  the  original  posi 
tion  of  S,  and  the  density  o-  =  Z>  cos  (A,  n)  over  the  rest  of 
the  surface.  There  would  have  been  on  S  a  surface  density 
<r  =  /lcos(A,  n^)  +  J2  cos  (A,  ?i2),  where  n^  and  n2  represent 
exterior  normals  to  the  regions  in  which  P  had  the  densi 
ties  pl  and  p2  respectively.  This  distribution  is  therefore 
equivalent  to  two  distributions  uniformly  polarized  in  the 
direction  of  h,  and  laid  together  so  as  to  have  the  common 
surface  S. 

If,  again,  the  density  of  P  had  been  given  by  the  expression 

P  =  PO  •/(*,  y,  «), 

where  p0  is  a  constant  and  /  an  analytic  function  of  the  space 
coordinates,  then,  if  P  had  been  displaced  parallel  to  the  x 
axis,  there  would  have  been,  (1)  a  region  common  to  P  and 
N  in  which  the  density  would  have  been 

po[f(x  -  h>  y> z)  -f(x>  y> «)]  or  -  p»h  •  D*f+ e 

where  e  is  an  infinitesimal  of  the  same  order  as  h,  (2)  a  region 
of  density  p0f(x  —  h,  y,  z)  where  P  extended  beyond  N,  and 
(3)  a  region  of  density  p0f(x,  y,  z)  where  N  extended  behind 
P.  If  the  limit  of  p0h  had  been  the  constant  A0,  and  if 
A-/(z,  y,  z)  had  been  denoted  by  A,  the  resulting  distri 
bution  would  have  had  a  surface  density  a-  =  A  cos  (x,  n)  over 
the  boundary  of  the  space  originally  occupied  by  N  and  P 
and  a  volume  density  p  =  —  DXA  inside  this  boundary.  This 
kind  of  distribution  is  called  a  non-uniform  polarization  of 
intensity  A,  the  direction  of  the  polarization  being  that 
of  the  x  axis.  We  know  from  Green's  Theorem  that  the 


ELECTROSTATICS.  193 

surface  integral  of  A  cos  (x,  n)  taken  over  any  closed  surface 
is  equal  to  the  volume  integral  of  +  D^A  taken  through  the 
space  bounded  by  the  surface,  so  that  the  whole  amount 
of  matter,  algebraically  considered,  in  the  distribution  just 
discussed  is  zero. 

If  such  a  distribution  as  this  were  placed  in  a  uniform  field 
of  force  of  intensity  F,  perpendicular  to  the  x  axis,  it  would 
encounter  a  couple  of  moment 

M=  F  CCx'<r-dS+  F  C  C  Cx.p.dr 

=  F  C  Cx  •  A  .  cos(ar,  n)dS  -  F  C  C  Cx  D^A  •  dr 


-  x 


Here,  again,  the  volume  integral  of  the  intensity  of  the  polar 
ization  is  a  measure  of  the  moment  of  the  couple  which 
would  be  exerted  upon  the  distribution,  if  it  were  placed  in  a 
uniform  field  of  unit  strength  perpendicular  to  the  x  axis. 
The  intensity  of  the  polarization  at  any  point  in  a  polarized 
distribution  has  been  called  the  moment  per  unit  volume  of  the 
distribution  at  the  point.  If  a  distribution  polarized  in  the 
manner  just  described  parallel  to  the  x  axis  were  placed 
in  a  uniform  field  (X0,  YQ9  Z0),  not  perpendicular  to  the  x 
axis,  it  would  experience  a  couple  the  components  of  which 
would  be 


If  p0V(x,  y,  z)  is  the  potential  function  at  (x,  y,  z)  due  to 
P  in  its  original  position,  the  potential  function  at  (or,  y,  z) 
due  to  N  and  P,  after  P  has  been  displaced  parallel  to  the 
axis  of  x  through  the  distance  /*,  is 

po  [  V(x  -  h,  y,  z)  -  V(x,  y,  «)],  or  -  pQh*DxV+e, 


194  ELECTROSTATICS. 

where  e  is  an  infinitesimal  of  the  same  order  as  h.  As  li  is 
decreased  and  p0  so  increased  that  pji  is  always  equal  to  AM 
the  potential  function  at  (x,  y,  z)  due  to  the  resulting  distri 
bution  becomes  —  A0  •  Dx  V.  Thus,  if  P  is  a  sphere  of  radius  «•, 
the  density  of  which  is  proportional  to  the  distance  from  its 
centre,  we  have  p  =  p0r,  V  =  7rp0(4#3  —  r3)  /3  if  r  <  a,  and 
V=TrpQa*/r  if  r  >  a.  The  polarization  in  the  resulting 
distribution  is  A0r,  where  A0  is  a  constant  to  be  chosen  at 
pleasure  ;  the  potential  function  has  the  value  irA^xr  within 
the  polarized  sphere,  and  -n-A^x  /r*  without  it  ;  the  moment 
of  the  sphere  is  7rA0a4. 

Imagine  six  coincident  bodies,  P1?  N^  P2,  N2,  P3,  N3,  of 
densities  p<>fi(x,y,z),  —  p0fi(x,  y,z),  p0f2(x,  ;>/,  z),  -/o0/2(*,y,«), 
Po/sO,  y>  «)>  -  Po/s(#»  2A  «)  respectively.  Imagine  P1?  P,,  P3 
displaced  through  distances  A  parallel,  respectively,  to  the  axes 
of  Xy  y,  and  z,  then  imagine  h  to  decrease  and  p0  to  increase 
in  such  a  way  that  p()A  is  always  equal  to  a  constant  M.  If 
J//i  (a;,  y,  «),  J/A  (a,  y,  s),  Jfjfa  (»,  y,  «)  be  denoted  by  A9  B,  and 
C  respectively,  the  resulting  distribution  has  a  surface  density 
a  —  A  cos  (#,  ri)-\-  B  cos  (y,  71)  +  (7  cos  («,  ?i)  on  the  boundary 
of  the  space  originally  occupied  by  the  six  bodies,  and  a  volume 
density  p  =  —  (DXA  +  ^?/5  4-  -A-^)  in  the  region  enclosed  by 
this  boundary.  ^4,  B,  C  are  usually  considered  to  be  the 
components  taken  parallel  to  the  coordinate  axes  of  a  vec 
tor,  /,  so  that  <r  —  /  cos  (n,  I)  and  p  =  —  (Divergence  of  /). 
The  whole  amount  of  matter  in  the  distribution  is  zero.  /  is 
called  the  polarization,  and  the  direction  of  I  at  any  point 
is  the  direction  at  that  point  of  the  polarization.  The  lines 
of  the  vector  /  are  defined  by  the  equations 


and  are  called  the  lines  of  polarization.  If  through  every 
point  of  a  curve,  s,  in  a  polarized  distribution,  we  draw  a 
line  of  polarization,  we  shall  get  (unless  s  is  itself  a  line  of 
polarization)  a  polarization  surface;  if  s  is  closed  and  the 
polarization  surface  tubular,  the  latter  is  called  a  tube  of 


ELECTROSTATICS  195 

polarization.  The  product  of  the  cross-section  of  a  very 
slender  tube  of  polarization  at  any  point,  and  the  value 
at  that  point  of  /,  is  sometimes  called  the  strength  of  the 
tube.  The  matter  in  a  slender  tube  of  polarization  con 
stitutes  a  polarized  filament.  If  the  vector  /  is  solenoidal. 
the  distribution  is  wholly  superficial,  and  the  strength 
of  every  tube  of  polarization  is  constant  throughout  its 
length.  Uniform  polarization  is  a  special  case  of  solenoidal 
polarization. 

It  is  evident  that  the  generally  polarized  distribution  just 
mentioned  may  be  regarded  as  formed  by  the  superposition 
of  three  distributions  polarized  parallel  to  the  axes  of  x,  //. 
and  z  respectively,  and  it  is  easy  to  see  that  a  uniformly 
polarized  distribution  in  a  uniform  field  (Xm  Yw  ZQ)  will  be 
acted  on  by  a  couple  the  components  of  which  are  the  prod 
ucts  of  the  volume  of  the  distribution  and  the  quantities 
BZ0  -  CYW  CX,  -  AZW  A  Y,  -  BX«. 


A  short,  extremely  slender,  right  prism,  uniformly  polar 
ized  in  the  direction  of  its  length,  forms  a  simple  kind  of 
polarized  element.  If  21  is  the  length  of  such  an  element,  q 
its  cross-section,  and  /  the  intensity  of  its  polarization,  2qII 
may  be  called  the  moment  of  the  element,  for  it  represents  the 
moment  of  the  couple  which  would  act  upon  the  element  if  it 
were  placed  perpendicularly  across  the  lines  of  a  unit  field. 
This  product  of  the  volume  of  the  element  and  /  we  may 
denote  by  H.  We  know  that  the  field  of  force  due  to  the 
element  is  mathematically  accounted  for  by  a  superficial 
negative  charge,  —  ql,  on  one  end  of  the  prism  and  an  equal 
positive  charge  on  the  other  end.  Let  Q  be  any  point  distant 
r±  from  the  negative  end  and  r2  from  the  positive  end  of  the 
axis  of  the  prism,  and  r  from  its  centre.  Let  (>•,  /)  be  the 
angle  between  the  direction  of  polarization  and  the  line  drawn 
from  the  centre  of  the  axis  to  Q ;  then,  since 

=  '/~  +  Z2  +  2  rl  •  cos  (r,  7)  and  r.r  =  >~  +  I'2  -  2  /•/ .  cos  (r,  /), 


196 


ELECTROSTATICS. 


the  value  at  Q  of  the  potential  function  due  to  the  element 
is     /lr-lr     or 


or 


cos  (r, 


The  limit,  Jf-cos(r,  I)/iP,  of  the  expression  just  found  is 
called  the  potential  function  due  to  a  uniformly  polarized 

element,  or  to  a  space 
doublet.  It  will  appear 
from  the  work  which  fol 
lows  that  a  similar  result 
might  have  been  obtained 
from  the  use  of  a  gener 
ally  polarized  element  of 
any  form.  The  lines  of 
force  due  to  a  polarized 
element  are  shown  in 
Fig.  53  ;  they  are  the 
same  as  the  external  lines 
of  force  in  Fig.  50. 

Before  we  attempt  to 
find  an  expression  for  the 
potential  function  due 
to  a  generally  polarized 
finite  distribution,  it  is 
well  to  notice  that  if 
the  vector  /  is  discon 
tinuous  at  any  surfaces, 
the  distribution  may  be 
FIG.  53.  considered  as  made  up 

of  a  number   of  contin 

uously  polarized  portions  abutting  at  these  surfaces  :  we 
may  confine  our  attention,  therefore,  to  continuously  polar 
ized  distributions.  If  a  given  distribution  of  this  kind  has 
the  volume  density  p'  in  a  region  T'  and  a  surface  density 
a-1  on  the  closed  surface  S'  which  bounds  T',  the  potential 
function  at  the  point  (x,  y,  z)  due  to  the  distribution  is 


ELECTROSTATICS.  197 


where  r  is  the  distance  from  the  point  (x',  y\  z')  in  the  volume 
or  surface  element  to  the  point  (x,  y,  z).  If  /'  is  the  value 
of  the  polarization  at  (x',  y',  z'),  and  if  we  substitute  the 
values  of  p'  and  <r'  in  terms  of  /'  and  its  components  A1,  £',  C', 
we  have 

r  r[A'  cos  (x',  n)  +  B'  cos  (y',  n)  +  C'  cos  (zf,  n)'] 

•— ^ z—  dr'. 


si- 

we  may  write 

-/// 


*,(£).  Sj*- 


*• 


'-DX>--^'     C  CA' 
*        -j  J  y 


with  similar  expressions  for  the  other  terms  of  the  triple  inte 
gral,  all  the  double  integrals  are  cancelled,  and  we  have 


r  = 


c'  •  D-  r    • 


. 


r 
P  cos  (r,  I')  J 


B'cosfy',  r)  +  C'cosfz'   r} 

-J- ii Ldr 


The  quantity  under  the  integral  signs  in  this  last  expression 


198  ELECTROSTATICS. 

is,  as  we  have  just  seen,  the  potential  function  due  to  an 
element  at  (x1,  y',  z'),  in  which  the  polarization  is  /'. 

If  the  polarization  is  solenoidal,  the  volume  integral  of  p' 
is  equal  to  zero  and  a  surface  integral  alone  remains. 

We  have  seen  that  a  polarized  distribution  is  completely 
defined  when  the  form  of  its  boundary,  S,  and  the  values  of 
the  components  of  the  vector  /  within  it  are  known,  and  that 
its  potential  function  has  the  same  value  (at  least  at  outside 
points)  as  that  due  to  an  ordinary  distribution  of  matter  made 
up  of  a  certain  volume  distribution  within  S  and  a  certain 
superficial  distribution  on  S.  What  we  usually  call  a  polar 
ized  distribution  is  supposed  to  be  quite  different,  however, 
in  its  physical  nature  from  this  ordinary  distribution,  which 
may  be  said  to  be  mathematically  equivalent  to  it.  A  simple 
illustration  will  make  the  character  of  this  difference  clear. 

If  a  number  of  small  cubes,  all  uniformly  polarized  parallel 
to  one  edge,  with  common  intensity  /,  were  placed  together, 
with  their  directions  of  polarization  parallel,  to  form  a  larger 
cube,  P,  superficial  distributions  of  equal  and  opposite  den 
sities  would  come  in  contact,  and  the  resulting  distribution 
would  appear  to  consist  only  of  a  positive  charge  uniformly 
spread  on  one  face  of  the  larger  cube  and  an  equal  negative 
charge  spread  uniformly  on  the  opposite  face.  That  is,  the 
potential  function,  at  outside  points,  due  to  P,  would  be  the 
same  as  that  due  to  an  indifferent  body,  P1,  of  the  same 
dimensions  as  P,  charged  with  a  superficial  distribution  of 
density  -f-  /  on  one  face  and  a  superficial  distribution  of  den 
sity  —  /  on  the  opposite  face.  If,  however,  we  define  the  force 
at  a  point  within  a  distribution  to  be  the  force  which  would 
urge  a  unit  mass  concentrated  at  the  point,  if  an  infinitesimal 
cavity  were  excavated  at  the  point  to  allow  of  its  introduc 
tion,  the  intensity  of  the  force  at  a  given  point  within  P' 
might  be  very  different  from  that  of  the  force  at  the  corre 
sponding  point  in  P.  The  first  would  be  due  merely  to  the 
surface  charges  already  mentioned,  whatever  the  shape  of 


ELECTROSTATICS.  199 

the  cavity,  while  if  an  excavation  were  made  in  P  by  remov 
ing  one  of  the  very  small  uniformly  polarized  cubes  of  which 
it  is  made  up,  the  surface  charges  on  the  adjacent  cubes  would 
appear,  and,  however  small  the  cavity  might  be,  these  would 
be  found  to  modify  the  force  very  appreciably.  We  must 
regard  such  polarized  distributions  as  occur  in  nature  as  made 
up  of  polarized  molecules,  so  that  if  any  portion  be  broken  off, 
across  the  lines  of  polarization,  from  a  body  in  which  the 
polarization  is  defined  by  the  vector  /,  each  portion  is  a 
polarized  distribution  defined  by  the  same  vector  as  before  at 
every  point,  so  that  a  surface  distribution  appears  on  each  of 
the  new  faces  formed  by  the  fracture.  Every  magnet  appears 
to  be  a  polarized  distribution  of  magnetic  matter,  and  prob 
lems  in  magnetism,  as  the  reader  who  has  some  knowledge 
of  magnetic  phenomena  will  see,  can  be  conveniently  attacked 
by  the  analysis  of  this  section. 


If  into  a  field  of  electric  force  a  conductor  or  a  mass  of 
dielectric  different  in  nature  from  that  which  it  displaces  be 
introduced,  the  field  becomes  changed  in  a  manner  completely 
explainable  on  the  assumption  that  the  conductor,  or  the 
dielectric,  has  become  electrically  polarized,  and  that  the 
surrounding  dielectrics  are  now  and  were  polarized.  Indeed, 
results  of  experiment  compel  us  to  assume  that  space  which 
seems  to  be  empty  of  ponderable  matter  is  still  occupied  by 
a  medium,  the  ether,  capable  of  transmitting  electrical  forces. 
We  must  assume,  also,  that  every  medium  with  which  we  are 
acquainted,  whether  it  be  solid,  liquid,  gaseous,  or  ethereal,  is 
susceptible  to  electrical  and  to  magnetic  forces,  so  that  if  a 
mass  of  any  isotropic  medium  be  placed  in  a  field  of  electric 
(or  magnetic)  force,  it  becomes  polarized  by  induction  in  such 
a  manner  that  the  direction  of  the  electric  (or  magnetic) 
polarization  coincides  at  every  point  with  the  direction  of 
the  resultant  electric  (or  magnetic)  force  due  to  all  the 
apparent  electrical  (or  magnetic)  matter  in  the  universe, 


200  ELECTROSTATICS. 

including  that  which  belongs  to  the  polarization  of  the 
medium  itself.  The  ratio  of  the  intensity  of  the  polariza 
tion  induced  at  any  point  of  a  medium  by  the  resultant  force 
at  the  point  is  called  the  susceptibility  of  the  medium  at  the 
point  under  the  given  circumstances.  Every  medium  has 
both  an  electrical  susceptibility  and  a  magnetic  susceptibility, 
and  these  may  be  represented  by  very  different  numbers. 
The  susceptibility  of  a  medium  to  magnetic  influences  often 
depends  upon  the  intensity  of  the  inducing  force;  we  may 
consider,  however,  that,  if  a  medium  is  homogeneous,  its  elec 
trical  susceptibility  (k)  has  the  same  value  throughout.  A 
medium  in  a  field  of  force  may  have  an  intrinsic  polarization 
as  well  as  the  polarization  induced  in  it  by  the  field.  A  steel 
magnet  in  the  earth's  field  illustrates  this  possibility. 

A  given  region  may  be  at  once  a  field  of  magnetic  force  and 
a  field  of  electric  force,  so  that  any  medium,  when  placed  in 
this  region,  becomes  both  magnetically  and  electrically  polar 
ized.  Since  the  two  polarizations  are  similar,  we  need  speak 
in  what  follows  only  of  one,  if  we  keep  in  mind  the  fact 
that  two  quite  independent  polarizations  may  coexist.  We 
shall  represent  susceptibility  by  k  and  inductivity  by  /n,  with 
the  understanding  that  different  numerical  values  must  be 
assigned  in  general  to  these  quantities  according  as  we  are 
dealing  with  electrical  or  magnetic  phenomena. 

In  the  most  general  case  of  either  electrical  or  magnetic 
polarization  we  may  imagine  that  an  isotropic  medium  has 
(1)  an  intrinsic  volume  charge  of  density  pw  where  p0  is  a 
scalar  point  function,  (2)  superficial  intrinsic  charges  over 
certain  surfaces,  and  (3)  an  intrinsic  polarization  70,  with 
components  Aw  Bw  C0,  which  may  or  may  not  be  everywhere 
continuous.  In  addition  to  this  it  has  (4)  an  induced  polar 
ization  which,  as  we  have  seen,  has  the  direction  of  the 
resultant  force  coming  from  all  the  apparent  charges  in 
existence,  including  those  which  come  from  the  intrinsic  and 
induced  polarization  of  the  medium  itself.  We  may  assume 
for  our  present  purposes  that  k  depends  upon  the  character 


ELECTROSTATICS.  201 

of  the  medium,  but  not  upon  the  intensity  of  the  resultant 
force  (Xj  Y,  Z).  The  whole  apparent  volume  density,  accord 
ing  to  the  statements  just  made,  is,  in  the  general  case, 

P  =  Po  ~  [JMo  +  AA  4-  A  CO 


and  this  is  equal,  according  to  Poisson's  Equation,  to 
-V2r/4ir,  or  to  (D^+D^Y+DtZ)/!*. 

If  we  denote  1  +  4  irk  by  /A,  this  equation  may  be  written  in 
several  interesting  forms,  and  may  be  regarded  as  a  general 
ized  Poisson's  Equation. 


=  4  *•[>„  -  (DXA,  4-  D,B0  +  AQ 

4  TT^o)  +   A  (pZ  +  4  7TQ  =  4 


+  A  [Z  4-  4  7r(A-Z  4-Q]  =  4  7r 

The  vector,  the  components  of  which  are  (/n-Y  4- 

4-  4  7T^0),  (/xZ  4-  4  7rC0),  or,  what  is  the  same  thing, 
(X+4«u4),  (r  +  47r5),  (Z  +  47rC),  where  .4,  ^,  (7  are  the 
components,  (&X4-  4,),  (A:F4-  A»)>  (^^  +  Q,  of  the  resultant 
polarization  arising  from  the  superposition  of  the  intrinsic 
and  the  induced  polarizations,  is  called  the  generalized  induc 
tion.  At  a  charged  surface,  the  sum  of  the  normal  compo 
nents  of  the  generalized  induction,  pointing  away  from  the 
surface  on  both  sides,  is  evidently  equal  to  4  TTO-O.  The  sum 
of  the  normal  components  of  the  force,  pointing  away  from 
the  surface  on  both  sides,  is  equal  to  47r<r,  while  every  tan 
gential  component  of  the  force  is  continuous  at  the  surface. 

If  in  a  homogeneous  medium  incapable  of  being  polarized 
inductively,  where  there  is  an  intrinsic  polarization  1^  (with 
components  A0,  Bw  <70),  but  no  intrinsic  body  or  surface 
charges  not  accounted  for  by  the  polarization,  the  polarization 
within  the  medium  coincides  everywhere  in  direction  with  the 


202  ELECTROSTATICS. 

force  due  to  all  the  active  matter  in  existence,  including  the 
polarization  masses,  and  if  the  intensities  of  the  polarization 
and  force  have  everywhere  the  constant  ratio  X,  the  divergence 
of  the  polarization  is  evidently  equal  to  A  times  the  diver 
gence  of  the  force,  and  Poisson's  Equation  becomes 

DXX  +  Dy  Y  +  DeZ  =  -  4  TrX  (DXX  +  Dv  Y  +DZZ), 

so  that  the  force  and  the  polarization  must  be  solenoidal. 
The  converse  of  this  theorem  is  evidently  not  true. 

Inductive  bodies  which  are  incapable  of  being  intrinsically 
polarized  are  sometimes  said  to  be  electrically  or  magneti 
cally  soft.  Most  isotropic  substances  seem  to  be  electri 
cally  soft.  Bodies  which  can  be  intrinsically  polarized,  but 
which  are  assumed  to  be  incapable  of  being  polarized  by 
induction,  are  sometimes  said  to  be  electrically  or  magneti 
cally  hard.  No  absolutely  hard  media  are  known  to  exist, 
but  the  magnetic  susceptibilities  of  some  permanent  magnets 
are  comparatively  small. 

The  generalized  Poisson's  Equation  becomes 
Dx  0*  X)  +  Dy  (p  Y)  +  Dg  0  ^)  =  4  7TPo 

in  the  case  of  a  body  which  has  no  intrinsic  polarization, 
and  the  generalized  induction  becomes  the  simple  vector 
(/*X,  /A  Y,  fjiZ)  discussed  in  the  last  section.  This  vector  coin 
cides  in  direction  with  the  resultant  force  at  every  point. 
At  a  charged  surface  which  also  separates  two  media  of 
different  inductivities,  the  tangential  components  of  the 
force  are  continuous,  but  the  product  of  the  tangential 
component  of  the  force  and  the  inductivity  is  clearly  not 
continuous.  The  normal  component  of  the  force  is  discon 
tinuous  by  4  TT  times  the  apparent  density  of  the  charge  on 
the  surface,  while  the  normal  components  of  the  induction 
are  discontinuous  by  4  TT  times  the  density  of  the  intrinsic 
charge  on  the  surface.  In  general,  therefore,  in  soft  media, 
neither  the  tangential  nor  the  normal  components  of  the 
induction  are  continuous,  but  the  directions  of  the  force  and 


ELECTROSTATICS.  203 

induction  coincide  with  each  other  close  to  the  surface  on 
both  sides  of  it. 

If  k  is  independent  of  the  intensity  of  the  resultant  force, 
the  volume  density,  -  \_Dx(kX)  +  Dy  (kY)  +  Dz  (kZ)~],  due  to 
the  induced  polarization  in  a  homogeneous  medium,  may  be 
written -&  (D.^  +  DVY  +  DZZ),  or  k-\'2V,  and  this  van 
ishes  at  all  points  where  there  are  no  intrinsic  body  charges. 
We  must,  therefore,  consider  homogeneous  dielectrics  about 
charged  bodies  to  be  solenoidally  polarized. 

If  a  mass,  M,  of  a  soft  homogeneous  medium  of  induc- 
tivity  filt  be  introduced  into  a  given  field  of  force  in  an 
indefinitely  extended  homogeneous  medium  of  inductivity  /u2 
which  contains  no  "real"  charges  at  a  finite  distance  from 
J/,  the  two  media  become  solenoidally  polarized,  there  is  an 
••apparent"  charge,  e',  at  the  surface  S,  of  J/,  and  the  poten 
tial  function  V  is  now  the  sum  of  the  given  potential  func 
tion  F0,  which  defined  the  field  when  the  place  occupied  by 
M  was  filled  with  medium  of  inductivity  ^  and  the  potential 
function  F7,  which  might  be  computed  from  the  expression 
Limit  2(rf«'/r),  since  it  is  equal  to  the  potential  function  in  a 
medium  of  unit  inductivity  due  to  a  real  charge,  e'.  If  n^  and 
n?  are  normals  to  S  drawn  respectively  into  and  out  of  J/,  we 
have  at  every  point  of  S, 

Mi  •  !>„,  V  +  H,  -  Dnt  V  =  0,  D,n  Va  +  A,  Fo  =  0, 
and,  if  A.  =  ^  /^ 

A  •  Dni  V  +  Dn2  V'  +  (\-  1)  Dni  r()  =  0, 

in  which  A  is  positive,  and,  since  F0  is  given,  the  last  term  of 
the  first  member  is  a  given  function,  f(x,  y,  z).  V  is  con 
tinuous  at  S,  it  is  harmonic  within  and  without  S,  and  it  van 
ishes  canonically  at  infinity ;  it  is  easy  to  show  that  all  these 
conditions  determine  V  (and,  therefore,  e')  uniquely.  For  if 
we  assume  that  two  different  functions  may  satisfy  the  condi 
tions,  and  if  we  denote  their  difference  by  u,  u  must  vanish 
canonically  at  infinity,  be  continuous  at  S.  and  satisfy  Laplace's 


204  ELECTROSTATICS. 

Equation  within  and  without  S.  On  S,  ^  •  Dniu  +  /x2  •  Dn^u  =  0. 
If,  now,  we  apply  [149]  to  u,  choosing  for  X  of  that  equation 
the  value  ^  in  the  space  within  S  and  the  value  //,2  in  the  space 
without  S)  it  is  evident  that  u  must  have  everywhere  the 
value  zero.  It  is  to  be  noted  that  changes  in  /x,  and  ^  which 
did  not  affect  their  ratio,  would  not  affect  V. 

If  the  strength  of  the  given  field  had  been  greater  in  the 
constant  ratio  m  than  it  was,  the  potential  function  V",  due  to 
the  apparent  charge  on  S}  would  have  been  larger  in  the  same 
ratio,  for  [149]  shows  that  the  difference  between  the  two 
functions  V"  and  mV  (both  of  which  vanish  canonically  at 
infinity,  are  continuous  at  S,  are  harmonic  within  and  without 
S,  and  at  /S  satisfy  an  equation  of  the  form 

X •  D,,,  V"  +  Dn2 V"  +  m. /(x,  y,  *)  =  0), 
is  identically  zero. 

If,  when  a  hard  body,  M,  solenoidally  polarized  intrinsi 
cally,  is  placed  in  a  field  of  force  in  a  homogeneous  soft 
medium  of  inductivity  unity,  the  directions  of  the  polariza 
tion  and  of  the  resultant  force  are  found  to  coincide  within 
M,  and,  if  the  ratio  of  the  intensities  of  these  vectors  is  equal 
at  every  point  of  M  to  the  constant  k,  the  potential  function 
within  and  without  M  is  the  same  as  if  M  were  a  homogene 
ous,  perfectly  soft  medium  of  susceptibility  k  and  induc 
tivity  1  +  4  irk,  polarized  inductively  by  the  original  field. 
To  prove  this  we  have  only  to  compare  the  properties  of  the 
potential  functions  in  the  two  cases.  Let  S  be  the  bounding 
surface  of  M}  let  ;/,  and  n2  represent  respectively  interior  and 
exterior  normals  to  S,  and  let  F0  be  the  potential  function 
due  to  the  original  field ;  then  F0  is  harmonic  within  S,  and 
on  S,  where  F0  is  continuous,  -DniF0  -4-  Z>M2F0  =  0.  Let  /'  be 
the  polarization  in  the  hard  body  M,  and  I"  the  polarization 
which  would  be  induced  if  S  were  filled  by  a  soft  medium  of 
inductivity  1  -f  birk,  and  let  Ff  and  F"  be  the  correspond 
ing  potential  functions.  t  The  components  of  F  and  7"  in  any 


ELECTROSTATICS.  205 

direction  at  any  point  within  S  are  respectively  equal  to 
—  k  times  the  derivatives,  at  the  point,  in  the  given  direc 
tion,  of  F0  +  V  and  F0  +  F".  The  density  o-'  of  the  real 
charge  of  S  in  the  case  of  the  hard  body  is  —  /'  •  cos  (»„  /') 
or  k  •  Dni  ( F0  +  F')  and  the  density  a"  of  the  charge  which 
would  be  induced  on  S,  if  M  were  displaced  by  the  soft 
medium,  is  -  /"  -  cos  (nl}  I")  or  £  •  Dni  (  F0  +  F").  On  S, 

An  ( fo  +v)+ A,  ( v*  +  n  =  -  4  IT*  .  AJ  ( *  o  +  n, 

and        (1  +  4  *k)  •  A,  (  Fn  +  F")  +  A,  (  FO  +  H  =  0, 
or  (1  +  4  TT*)  -  A!  r  +  A2  F  =  -  4  TT£  •  At  *w 

and          (1  +  4  TrA')  -  At  F"  +  A2  r"  =  -  4  vk  •  Dni  F0. 

If,  now,  u  =  V  —  F",  u  is  harmonic  within  and  without  S, 
it  vanishes  canonically  at  infinity,  and  it  is  continuous  on  S, 
where  (1+4  TT&)  •  A,M  +  AaM  =  °-  Jt  is  eas3r  to  ProveJ  witn 
the  help  of  [149],  that  under  these  circumstances  u  must  be 
identically  equal  to  zero,  so  that  F'  and  F"  are  identically 
equal. 

We  know  from  work  done  earlier  in  this  section  that  when 
a  uniformly  polarized  sphere  is  placed  in  a  uniform  field  of 
force  of  intensity  XQ,  so  that  the  direction  of  the  polarization 
and  this  field  coincide,  the  resultant  field  within  the  sphere  is 
a  uniform  field  of  intensity  X$  —  4^/3  in  the  direction  of 
the  polarization,  and  that  the  ratio  of  the  polarization  and  the 
force  is  the  constant  k  =  3I/  (3  X(l  —  4  TT/),  or  3  /4  TT  (x  —  1), 
so  that  /=3JT0[(1  +47rA-)-l]/47r[(l  +4^  +  2].  We 
infer  from  this  that  if  a  sphere  of  soft  medium  of  inductivity 
/x  were  placed  in  a  uniform  field  of  force  of  intensity  Xn  in 
a  soft  medium  of  unit  inductivity,  the  sphere  would  become 
uniformly  polarized  to  intensity  3 XQ(^  —  l)/47r(/x  +  2)  and 
that  the  uniform  field  inside  the  sphere  would  have  the 
direction  of  the  outside  field  and  the  intensity  3JT0/(/x  +  2). 
Since  in  such  a  case  as  this  the  ratio  of  the  inductivities  of 
the  inner  and  outer  media  is  alone  important,  we  may  say 


206  ELECTKOSTATICS. 

that,  if  a  soft  sphere  of  inductivity  ^  were  placed  in  a 
uniform  field  of  intensity  JT0  in  a  soft  medium  of  inductivity 
/u,2,  the  sphere  would  become  uniformly  polarized  to  intensity 

3  X0  0x^-1)  74^/^4- 2), 
or 

and  that  the  intensity  of  the  uniform  resultant  field  within 
the  sphere  would  be  3X0/(/u,ly//x2  +  2)  or  3/x2JF0/(/z1  -f  2/x2). 
That  part,  JT0(/A,  —  /u,2)  /(^  H-  2/x2),  of  the  field  within  the 
sphere  which  is  due  to  the  polarization  alone,  is  negative,  if 
/A!  >  fj.2,  and  is  then  called  the  self -depolarizing  force. 

If  we  note  that  in  the  analysis  accompanying  Figs.  51  and 
52,  x  was  defined  by  the  equation  -3T0  =  47r/x/3  and  that 

consequently  x  =  OiM  +  2)/(/*iM  -  !)>  we  maJ  aPp!y  to 
our  present  subject  all  the  work  there  done.  These  figures 
represent  the  lines  of  force  for  the  cases  /x1//x2  =  <x>  and 
/ji,l/fjL2  =  %  respectively;  the  first  corresponds  to  a  perfect 
conductor  in  a  uniform  electric  field,  or  (approximately)  to  a 
sphere  of  very  soft  iron  in  a  uniform  magnetic  field  in  air. 
The  theory  of  the  polarization  by  induction  of  a  soft  sphere 
in  a  uniform  field  was  first  given  by  Lord  Kelvin,  and  this 
theory,  with  diagrams  for  /^//AO  —  2.8  and  1*1/1*2  =  0.48,  may 
be  found  in  his  Reprint  of  Papers  on  Electrostatics  and  Mag 
netism.  Very  interesting  figures,  drawn  for  equal  intervals 
of  the  function  m  and  corresponding  to  /Xi//x2  =  3,  /u,i//x2  =  oo, 
are  given  on  pages  373  and  374  in  Professor  Webster's  Theory 
of  Electricity  and  Magnetism. 

If  an  ellipsoid,  made  of  inductively  hard  material  and  uni 
formly  polarized  [/—  (A,  B,  (7)],  be  placed  in  a  uniform  field 
of  force  (Xw  Yw  Z0),  the  resultant  field  within  the  ellipsoid 
will  evidently  be  uniform  and  its  components  will  be 

XQ  -  2  7rabc.AKw  Y0-2  irabcBLM  ZQ-2  vabcCMn ; 

the  directions  of  the  polarization  and  the  field  will  not  agree, 
however,  unless  these  components  are  as  A  :  B :  C,  which  will 
be  the  case  if 


ELECTROSTATICS.  207 

A  =  kX0/  (1  +  2  irabcklQ, 
B  =  kY0/(l+2  7rabckL0), 
C  =  kZ0/(l  +  2  irdbckM^, 

where  k  is  any  constant.  If  the  intrinsic  polarization  of  the 
ellipsoid  satisfies  these  conditions,  the  ratio  of  the  intensities 
of  the  polarization  and  the  field  is  k.  We  infer  from  this 
that  if  a  homogeneous  ellipsoid  of  inductively  soft  material 
of  susceptibility  k  be  placed  in  a  uniform  field  of  force  in  a 
medium  of  unit  inductivity,  the  ellipsoid  will  become  uni 
formly  polarized  and  that  the  components  of  the  polarization 
will  have  the  values  just  given.  The  resultant  field  within 
the  ellipsoid  will  have  the  components 

X0/(l+2irabckK0),  Y0/(l  +  2irabckL0),  Z0/(l +  2'irabckM0) 
and  the  self-depolarizing  force  the  components 

-  2  irabcKfiA,  -  2  irabcL{,B,  -  2  TrabcM0C. 

These  results  may  be  expressed  in  terms  of  the  inductivity  p. 
of  the  ellipsoid  by  writing  (/w,  —  l)/47r  for  k,  and  if,  then, 
the  ratio  ^  /  ^  be  substituted  for  /x,  the  formulas  will  corre 
spond  to  the  case  of  a  soft  homogeneous  ellipsoid  of  induc 
tivity  /M!  in  a  field  of  force  in  a  homogeneous  medium  of 
inductivity  /u,,. 

If  we  remember  the  expression  already  found  for  the 
moment  of  the  couple  which  a  uniform  field  exerts  upon  a 
uniformly  polarized  distribution  in  it,  we  shall  see  that  in 
the  present  case  the  components  of  this  couple  are 

4  TTCibc  (BZ0  -  C  F0)  /  3,  4  Trabc  (  CXQ  -  AZ0)  /  3, 

and         4  nabc  (A  Y0  -  BX^  /  3, 

or 

8  irWbWk*  Y»Z()  (M0  -  LQ)  /3  (1  +  2  TrabckMQ)  (1  +  2  7rabckL0), 
8  7r-«2ft2c2/t%X0(JT0  -  MQ)  /3  (1  +  2  vabckKJ)  (1+ 
8  ir2a2b2c*k*X0  F0  (L0  -  JC0)  /  3  (1  +  2  TrabckL^  (1  +  2 


208  ELECTROSTATICS. 

and  it  is  to  be  noted  that,  according  to  the  reasoning  of  page 
122,  if  a  >  b  >  c,  KQ  <  L0  <  MQ.  If  the  lines  of  the  field  in 
the  air  make  an  acute  angle  with  the  plane  of  xz  and  are 
perpendicular  to  the  z  axis,  Z0  =  0  and  XQ  and  Y0  are  positive, 
the  axis  of  the  couple  is  the  axis  of  z,  the  moment  is  positive 
(even  for  such  negative  values  of  k  as  occur  in  nature),  and 
the  ellipsoid  tends  to  turn  so  that  its  long  axis  shall  have 
the  direction  of  the  field. 


For  perfectly  hard,  intrinsically  polarized  bodies  the  gen 
eralized  Poisson's  Equation  becomes 


and  if  p0  =  0,  as  in  the  case  of  a  hard  magnet,  the  induction 


is  solenoidal.  Unless  the  intrinsic  polarization  happens  to 
have  the  same  direction  as  the  resultant  force,  or  vanishes, 
the  lines  of  force  and  the  lines  of  induction  do  not  coincide. 
In  some  cases,  the  directions  of  the  force  and  of  the  polariza 
tion  are  exactly  opposed,  and  the  lines  of  force  and  of  induc 
tion  are  opposite  in  direction.  Outside  a  hard  magnet,  where 
the  intrinsic  polarization  is  nothing,  the  lines  of  force  and  of 
induction  are  identical.  At  the  surface  of  the  magnet,  where 
there  is  no  intrinsic  charge  except  that  which  belongs  to  the 
polarization,  the  normal  component  of  the  force  is  discontin 
uous,  while  the  normal  component  of  the  induction  is  con 
tinuous.  It  is  convenient,  therefore,  to  regard  the  lines  of 
induction  as  closed  curves.  The  lines  in  Fig.  50  represent 
both  lines  of  force  and  lines  of  induction,  but  it  is  to  be 
noticed  that  inside  the  sphere,  where  X=  —  4  7T//3,  the  direc 
tion  of  the  lines  of  force  is  from  right  to  left,  while  the  direc 
tion  of  the  lines  of  induction  (since  /=  4-  8  ir//3)  is  from 
left  to  right.  The  straight  line  of  force  through  the  centre 
of  the  sphere  is  discontinuous  in  direction  at  the  surface, 
while  the  corresponding  line  of  induction  is  continuous.  The 


ELECTROSTATICS.  209 

tangential  components  of  the  force  are  continuous  at  the  sur 
face  of  a  magnet,  those  of  the  induction  discontinuous.  The 
normal  component  of  the  induction  just  within  the  surface  is 

(X  +  4  TT  AO)  cos  (x,  n)  +  (  Y  +  4  TT  S0)  cos  (y,  ») 

+  (Z  +  4  TT  C0)  cos  (2,  w), 

or  the  normal  component  of  the  force  plus  4ir  times  the 
density  on  the  surface  belonging  to  the  intrinsic  polarization. 
If  we  make  a  small  cavity  inside  a  generally  polarized  hard 
body,  the  force  at  any  point  of  the  cavity  is  the  original  value 
of  F  (that  is,  the  negative  of  the  gradient  of  the  potential  func 
tion  at  the  point),  minus  the  contribution  due  to  the  volume 
charge  removed,  plus  the  force  due  to  the  new  surface  charges 
which  appear  on  the  walls  of  the  cavity.  If  the  volume  of 
the  cavity  be  made  smaller  and  smaller,  the  contribution  due 
to  the  volume  charge  removed  can  be  made  as  small  as  we 
like,  while  the  effect  of  the  surface  charges  may  remain  finite. 
Let  the  cavity  be  of  the  form  of  a  piece  of  a  slender  tube  of 
polarization  lying  between  two  near  orthogonal  surfaces.  In 
this  case  there  will  be  surface  charges  on  the  ends  of  the 
cavity,  but  none  on  the  side  walls.  These  charges,  of  density 
±  /,  will  be  such  as  to  drive  a  particle  of  positive  matter  in 
the  centre  of  the  cavity  to  that  end  of  the  cavity  towards 
which  the  polarization  is  directed.  If  we  reflect  that  a  surface 
distribution  of  finite  sii«,  which  has  at  the  point  P  the  density 
«r,  repels  a  unit  point  charge  infinitely  near  P  with  a  force  of 
2  TTO-,  but  that  an  element  of  the  surface  at  P,  infinitely  small 
with  respect  to  the  distance  of  a  point  charge  from  P,  has  no 
perceptible  effect  upon  this  point  charge,  it  will  be  easy  to 
see  that,  if  the  cross-section  of  the  cavity  is  infinitely  small 
compared  with  its  length,  the  force  due  to  the  surface  charges 
on  the  ends  approaches  zero  as  the  whole  cavity  is  made 
smaller,  and  the  force  at  the  centre  of  the  cavity  is  F.  If, 
however,  the  length  of  the  cavity  is  infinitely  small  compared 
with  its  cross-section,  the  force  due  to  the  charges  on  the 
ends  is  4  TTO-.  or  4  TT/,  so  that  the  whole  force  is  F  +  4  TT/,  or 


210  ELECTROSTATICS. 

the  induction  at  the  point  before  the  cavity  was  cut.  If  the 
infinitely  small  cavity  were  spherical,  the  force  at  its  centre 
would  be  F  +  irl. 


It  is  to  be  carefully  noticed  that  we  have  no  means  of 
determining  an  absolute  inductivity  for  any  medium,  but  only 
the  ratio  of  the  inductivity  of  the  medium  to  the  inductivity  of 
some  other  medium  taken  as  a  standard.  The  unit  quantity 
of  electricity  is  defined  to  be  that  quantity  which  concentrated 
at  a  point  at  a  distance  of  one  centimetre  from  an  equal 
quantity  would  repel  it  with  a  force  of  one  dyne,  when  the 
dielectric  is  the  ether.  In  any  other  homogeneous  dielectric 
of  inductivity  /x  times  that  of  the  ether,  e  of  these  units  of 
electricity  concentrated  at  each  of  two  points  distant  r  centi 
metres  from  each  other  would  repel  each  other  with  a  force 
of  e^/pr*  dynes,  so  that,  if  this  medium  had  been  used  as  a 
standard,  the  unit  of  electricity  would  have  been  larger  in  the 
ratio  of  v/x  to  1  than  it  now  is.  If  a  charged  conductor  is 
enveloped  by  an  infinite,  homogeneous  dielectric,  we  may  assume 
the  apparent  charge  on  it  to  be  its  real  charge  and  neglect  the 
polarization  of  the  dielectric  (and  we  do  this  when  the  dielec 
tric  is  the  standard  substance  which  we  assume  to  have  unit 
inductivity,  and  hence  no  susceptibility)  ;  or  we  may  suppose 
the  dielectric  to  be  polarized,  and  consider  the  apparent 
charge  to  be  the  algebraic  sum  of  the  real  charge  on  the 
conductor  and  the  charge  belonging  to  the  polarization  induced 
on  the  dielectric.  This  we  do  when  the  dielectric  is  not  the 
standard  substance,  assigning  to  it  a  susceptibility  based  on 
that  of  the  standard  substance  which  is  the  zero  of  our  scale. 
A  simple  illustration  will  tend  to  make  the  rather  complex 
relations  which  would  attend  a  change  in  the  choice  of  a 
standard  substance  more  intelligible. 

A  spherical  conductor  of  radius  a  has  a  charge,  E,  and  is 
surrounded  by  three  spherical  shells  of  homogeneous  dielectric, 
concentric  with  it,  the  last  reaching  to  infinity.  The  radii  of 
the  spherical  surfaces  which  separate  the  first  (inner)  dielectric 


ELECTROSTATICS.  211 

from  the  second,  and  the  second  from  the  third,  are  b  and 
c  respective!}-,  and  the  inductivities  of  the  dielectrics,  referred 
to  some  standard  substance,  are  /nl5  /x.2,  and  /*3,  or  1+4  irkly 
1+4  TJ-A-O,  and  1  +  4  7rA-3.  If  we  apply  Gauss's  Theorem  suc 
cessively  to  spherical  surfaces  concentric  with  the  conductor 
and  lying  in  the  first,  second,  and  third  media,  we  learn  that 
the  force  (—  DrV)  at  a  distance  T  from  the  centre  has,  in  the 
three  media,  the  values  E  /  '/^r2,  E  '/  ^2r2,  E  /  >3r2.  The  con 
ductor  acts  like  a  medium  of  infinite  susceptibility.  The 
induced  polarizations  in  the  three  media  are  directed  radially 
outward,  and  their  intensities  are  ^E/^r2,  kc.E/p.c.r*,  kzE/w\ 
The  densities  of  the  apparent  charges  at  the  surface  Sl  of  the 
conductor  and  at  the  surfaces  of  separation  Sz,  Ss  of  the 
dielectrics,  regarded  as  manifestations  of  the  polarizations,  are 

on  S» 

o-2  =    if*     —    2(j..2     =       fil  —  /x2/47r^2/x1/A2  on  S& 
and  o-'  =  kEL*  -  kE          =  E       -  i*^2         on  s 


These  same  densities  might  also  be  found  by  the  aid  of  the 
ordinary  characteristic  equation  of  the  potential  function  at 
an  apparently  charged  surface,  Dn.V  +  Dn,V—  —  47ro-'. 

If  instead  of  using  the  old  standard  we  make  the  outer 
dielectric  of  this  problem  the  standard,  the  unit  of  electrical 
quantity  will  be  larger  in  the  ratio  of  V^  to  1  than  it  was 
before,  and  the  old  charge  on  the  condenser  will  be  E/^JL- 
expressed  in  the  new  units.  The  strength  of  a  field  at  any 
point  being  the  force  in  dynes  which  would  be  experienced 
by  a  unit  of  positive  matter  placed  at  the  point,  the  number 
which  expresses  the  strength  of  a  given  field  in  the  new  units 
is  Vj^  times  the  number  which  expresses  the  strength  of  the 
same  field  in  the  old  units.  The  inductivities  of  the  three 
dielectrics  are  now  /M//^,  /x2//x3,  1,  and  their  susceptibilities 

are 

3  or  (A-!  -  A-3)/(l  +  47rA-3), 

x3  or  (A-o  —  A-3)  '(1  +  47rA-3),  and  0. 


212  ELECTROSTATICS. 

The  strengths  of  the  fields  in  the  three  dielectrics  are 


the  intensities  of  the  polarizations  are 

E(^  -  Ms)/4  T  V^  nt*9    ^(/*2  -  /x3)/4  TT  V^  /x^2,  0  ; 
and  the  apparent  charges  on  Sl9  S2,  Ss  have  the  densities 

E  V/T3/4  Tra2,*!, 
and  ^  (/x2  — 

The  sum  of  the  apparent  charges  on  Si,  $2,  $3  is  now  .Z?/  V/x3, 
the  real  charge  on  the  conductor  expressed  in  the  new  units, 
and  the  sum  of  the  induced  charges  is  zero.  In  the  case  first 
treated,  where  the  outer  medium  was  supposed  to  be  polarizable, 
the  sum  of  the  apparent  charges  on  Si,  Sz,  S3  was  E  /  ^  and 
this,  being  expressed  in  the  old  units,  is  equivalent  to  i?/  V/x3 
in  the  new.  The  sum  of  the  induced  charges  was  the  difference 
between  E  /  /x3  and  E  or  E(l  —  ju,3)//>t3;  in  this  case,  however, 
we  must  imagine  the  outer  surface  "at  infinity"  of  the  outer 
medium  to  have  an  induced  charge  in  total  amount  equal 
to  the  integral  of  the  normal  component  of  the  polarization 
(k3E/p.3r2)  over  the  surface,  or  4  irk^E  /  ^  and  this  is  equal 
to  E(fjL3  —  l)//n3,  so  that  here,  again,  the  whole  amount  of  the 
induced  charge  is,  of  course,  zero.  It  is  to  be  noted  that  this 
finite  charge  at  infinity  does  not  affect  the  electrical  field  in 
any  way.  We  have  seen  that  when  the  outer  medium  is 
taken  as  a  standard  the  inner  medium  has  a  susceptibility 
(^!  —  /x3)/47r/x3,  and  this  is  sometimes  called  the  susceptibility 
of  a  medium  of  inductivity  ^  with  respect  to  a  medium  of 
iiiductivity  /x3.  No  medium  has  yet  been  found  to  be  less 
electrically  susceptible  than  the  ether.  Some  bodies  are  less 
magnetically  susceptible  than  the  ether,  so  that  their  suscepti 
bilities  are  negative  on  the  usual  scale.  These  bodies  are 
said  to  be  diamagnetic. 

If  a  body  of  inductivity  /*!,  bounded  by  the  surface  S,  is 
placed  in  a  large  mass  of  a  medium  of  inductivity  /x2,  the  outer 


ELECTROSTATICS.  213 

surface  of  which  is  so  far  removed  from  the  place  of  observa 
tion  that  the  apparent  charge  on  it  contributes  little  to  the 
field  of  force,  the  fact  that  the  outer  medium  is  really  polarized 
may  be  lost  sight  of ;  and  if  we  attribute  the  apparent  charge 
on  S  wholly  to  the  polarization  of  the  inner  medium,  instead 
of  regarding  it  as  the  difference  between  the  charge  of  one 
sign  due  to  the  polarization  of  the  inner  medium,  and  the 
charge  of  the  opposite  sign  due  to  the  polarization  of  the 
outer  medium,  the  apparent  susceptibility  of  this  medium 
will  be  (/A!  —  p^)/±7rfjL<,.  If  fj.2  is  greater  than  /xx,  this  will  be 
negative  and  the  inner  medium  will  seem  to  be  polarized  in 
a  direction  opposite  to  that  of  the  force. 


If  in  any  given  case  the  direction  of  the  vector  /  is  every 
where  perpendicular  to  the  direction  of  its  curl,  it  is  possible 
to  cut  a  polarized  distribution  by  a  set  of  surfaces,  u  =  c, 
everywhere  normal  to  the  line  of  polarization.  If  surfaces  of 
this  family  be  drawn  for  small  constant  differences,  Aw,  of  the 
scalar  point  function  u,  the  distribution  will  be  divided  into 
shells,  each  of  which  is  polarized  normally  to  its  surface.  If 
AM  is  the  thickness  of  one  of  these  shells  at  a  given  point  and 
/0  the  average  intensity  of  polarization  on  a  line  of  polariza 
tion  drawn  through  the  shell  at  the  point,  J0A;i  is  called  the 
strength  of  the  shell  at  the  point.  Since  DHu  =  hu,  the  value 
of  the  gradient  of  ?/,  the  strength  of  a  shell  of  infinitesimal 
thickness  can  be  written  I-du/hu.  A  shell  is  said  to  be 
simple  if  I/ku  has  the  same  numerical  value  all  over  it; 
otherwise  the  shell  is  said  to  be  complex. 

If  A,  B,  C  are  the  intensities  of  the  components  of  the 
vector  /,  the  fact  that  the  lines  of  /  coincide  with  the  nor 
mals  to  the  surface  u  =  c  gives  the  scalar  equations 

A/I=  Dxit/hu,      B/I=  Dyu/ku,      C/I=  I)M/h^ 
and  with  the  help  of  these  the  vector 

\D,C  -  DSB,  DZA  -  DXC,  DXB  -  DyA^ 


214  ELECTROSTATICS. 

which  is  the  curl  of  /,  may  be  written  in  the  form 
\_Dzu  •  D!t  (  T  I  //„)  -  DyU  • 


Dvu  •  Dx(I/hH)  -  Dxn  -  - 

If,  now,  the  scalar  quantity  I/hu  has  the  same  numerical 
value  over  every  surface  of  constant  u,  it  must  be,  if  not 
everywhere  constant,  a  function  of  n  only,  so  that 

Dx(I/hH)  :  A,"  =  Dv(I/hu)  :  Dyu  =  Dz  (///,.„)  :  Dgu, 

and  if  these  relations  are  satisfied,  the  components  of  the  curl 
of  /  vanish,  and  the  polarization  is  lamellar.  Every  lamel- 
larly  polarized  distribution  may  be  divided  up  into  simple 


FIG.  54. 

polarized  shells ;  if  the  polarization  is  not  lamellar,  but  if  the 
directions  of  this  vector  and  its  curl  are  everywhere  perpen 
dicular  to  each  other,  the  distribution,  as  we  have  seen,  may 
be  divided  up  into  shells,  but  these  will  not  be  simple. 

The  potential  function  due  to  a  polarized  element  of  moment 
M  has  at  a  point,  P,  distant  r  from  the  element,  the  value 
M cosa/r2,  where  a  is  the  angle  which  a  line  drawn  from  the 
element  to  P  makes  with  the  direction  of  polarization  in  the 
element.  If  a  very  thin  simple  shell  be  divided  up  into  ele 
ments  of  length  equal  to  the  thickness  (A??)  of  the  shell  and 
of  cross-section  equal  to  an  element  (A$)  of  one  surface  of 
the  shell,  the  moment  of  each  element  is  A$-/A?i,  and  if 
,  the  constant  strength  of  the  shell,  be  denoted  by  <£,  the 


ELECTROSTATICS.  215 

potential  function  due  to  the  shell  has  at  any  point,  P,  the  value 
limit  <£^  cos(r,  n)±S/r*  =  <£<o.  where  to  is  the  solid  angle 

subtended  at  P  by  the  boundary  of  the  shell.  This  value  is 
positive  if,  in  looking  out  from  the  vertex  P  within  the  conical 
surface  which  passes  through  the  boundary  of  the  shell,  one 
sees  the  positive  side  of  the  shell.  If,  while  the  strength  of 
the  shell  is  unchanged  and  the  boundary  fixed,  the  shell  itself 
be  imagined  deformed  in  any  way,  the  value  at  P  of  the  poten 
tial  function  due  to  the  shell  will  be  unchanged  so  long  as  P 
is  on  the  same  side  of  the  shell.  The  potential  function  due  to 
a  closed  simple  shell  of  any  form  is  zero  at  every  outside  point 
and  ±  4  TT<£  at  every  inside  point,  where  the  positive  sign  is  to 
be  used  if  the  positive  side  of  the  shell  is  turned  inwards. 

If  P  and  P'  are  two  points  close  to  each  other  on  opposite 
sides  of  a  simple,  very  thin  shell,  S,  of  strength  <l>,  and  if 
VP  and  VP.  are  the  values  of  the  potential  function  at  P  and  P', 
due  to  S,  we  may  imagine  the  shell  closed  by  an  additional 
shell  also  of  strength  3>  which  shall  add  to  the  potential  func 
tions  at  each  of  the  near  points  P  and  P'  the  quantity  x.  If 
P  is  within  the  closed  shell,  P1  will  be  outside,  so  that 

V  +  x  =  0,      V  4  x  =  ±  4  irfc,  or    V  -  V  =  ±  4  ^. 

The  potential  function  due  to  an  infinitely  thin,  open  or  closed, 
simple  polarized  shell  is,  therefore,  discontinuous  at  the  shell 
by  ±  4  TT  times  the  strength  of  the  shell. 

The  potential  energy  of  a  magnetic  north  pole  of  strength  m 
at  a  point,  P,  near  a  simple,  finite  magnetic  shell  is  ±  M<£U>, 
and  if  P  is  on  the  positive  side  of  the  shell,  ?>t<£<o  ergs  will  be 
done  by  the  field  on  the  pole  if  it  be  carried  to  infinity  by  any 
path.  If  the  pole  be  carried  around  the  edge  of  the  shell  from 
a  point  very  near  the  shell  on  the  positive  side  to  a  point  very 
near  the  first  but  on  the  negative  side,  the  work  done  on  the 
pole  by  the  field  will  l>e  4:inn&  ergs. 

In  general, 

cos(/>,  '')   . 


216  ELECTROSTATICS. 

where  r  is  the  distance  from  dS  to  P,  and  n  is  the  normal  to 
the  shell  on  the  positive  side.  If  the  directions  of  both  r  and 
n  were  considered  reversed,  the  value  of  the  integral  would  be 
unchanged,  but  it  would  then  more  clearly  represent  the  sur 
face  integral,  taken  over  the  shell,  of  the  normal  component 
towards  the  negative  side  of  the  shell  of  the  force  due  to  mag 
netic  pole  at  P.  If,  instead  of  a  single  pole  at  P,  there  is  any 
collection  of  poles  at  different  points  or,  indeed,  any  magnetic 
distribution,  M,  the  mutual  potential  energy  of  the  shell  and 
this  distribution  is  equal  to  <f>  times  the  flux  of  magnetic  force 
due  to  M  in  the  negative  direction  through  the  shell. 

A  simple  magnetic  shell   in  a  magnetic  field,  Hw  due  to 
matter  outside  the  shell  tends  to  move  so  as  to  decrease  the 


FIG.  55. 

mutual  potential  energy  of  the  shell  and  the  field,  and  this 
quantity,  as  we  have  just  seen,  is  equal  to  the  negative  of  the 
product  of  the  strength  of  the  shell  and  the  number  N  of 
lines  (unit  tubes)  of  force  due  to  the  field  which  cross  the 
shell  in  the  positive  direction.  The  shell,  therefore,  tends  to 
move  so  as  to  make  N  as  great  as  possible.  If  the  shell  be 
displaced  parallel  to  itself  through  a  very  short  distance,  du, 
in  any  direction,  the  limit  of  the  ratio  of  the  loss  of  energy 
(+®-dN)  caused  by  the  displacement  to  du  (i.e.,  4>  •  DUN) 
measures  the  force  U,  which  tends  to  move  the  shell  in  this 
direction. 

If  we  suppose  that  the  shell  in  being  displaced  does  not 
encounter  any  of  the  magnetic  matter  which  gives  rise  to  the 
field,  Zf0  will  be  a  solenoidal  vector  within  the  cylinder 


ELECTROSTATICS.  217 

generated  by  the  shell,  so  that  the  integral  of  the  normal  out 
ward  component  of  HQ  taken  over  the  surface  of  the  cylinder 
will  be  zero.  The  shell  in  its  initial  and  final  positions  forms 
the  ends  of  the  cylinder,  and  these  together  contribute  dN 
to  the  surface  integral,  so  that  the  convex  surface  must  con 
tribute  —  dN.  If  ds  is  an  element,  measured  in  the  positive 
direction  about  the  shell,  of  the  curve  which  bounds  the  shell 
in  its  original  position,  and  if  dS  is  the  element  of  the  convex 
surface  of  the  cylinder  generated  by  ds, 

dS  =  ds  •  du  •  sin  (du,  ds)  ; 

the  magnetic  induction  through  this  element  due  to  the 
magnetic  matter  outside  the  shell  is 

HQ  •  cos  (n,  7/o)  •  sin  (du,  ds)  •  du  •  ds, 

and  this  integrated  with  respect  to  s  is  equal  to  —  dN,  or 
to  —  Udu/&.  Therefore, 

V  =  —  &  (  HQ-  cos  (n,  HQ)  •  sin  (du,  ds)  •  ds, 

and  the  component  in  any  direction  (u)  of  the  whole  force  on 
the  shell  may  be  expressed  as  a  line  integral  taken  around  the 
curve  which  bounds  the  shell.  The  integrand  vanishes  at  any 
point  where  u  is  parallel  to  HQ  or  to  ds,  but  if  at  any  point  u 
happens  to  be  perpendicular  to  the  plane  of  HQ  and  ds,  the 
integrand  becomes  HQ  sin  (Hw  ds),  the  component  of  the  field 
perpendicular  to  ds.  If,  with  this  fact  in  mind,  we  choose  at 
every  point  on  the  curve  a  direction,  p,  perpendicular  to  the 
plane  of  If^  and  ds,  so  that 

cos  (p,  ds)  =  0  and  cos  (p,  If0)  =  0, 

and  remember  that  cos  (n,  ds)  =  0,  cos  (u,  n)  =  0,  we  may 
easily  prove  that 

HQ  •  cos  (n,  H0)  •  sin  (du,  ds)  =  HQ  •  sin  (HQ,  ds)  -  cos  (p,  u). 

This  shows  that  U  may  be  mathematically  accounted  for  by 
assuming  that  every  element  of  the  curvilinear  boundary  of 
the  shell  is  urged  in  a  direction  perpendicular  to  the  field  and 


218  ELECTROSTATICS. 

to  the  element,  by  a  force  numerically  equal  to  the  product 
of  the  length  of  the  element,  the  strength  of  the  shell,  and 
the  component  perpendicular  to  the  element  of  the  field,  JT0. 

If  the  field  is  due  to  a  single  magnetic  pole  of  strength  m 
at  a  point,  P,  distant  r  from  ds,  the  force  on  the  element 
would  be  w<3>-  sin(r,  ds)  -ds/r2,  and  the  force  exerted  by  the 
shell  on  the  pole  would  be  accounted  for  by  assuming  that 
every  element,  ds,  of  the  boundary  of  the  shell  contributed 
an  elementary  component,  w<J>-sin(>,  ds)ds/r2,  in  a  direction 
perpendicular  to  the  plane  of  P  and  ds. 

VECTOR  POTENTIAL  FUNCTIONS  OF  THE  INDUCTION. 

Every  vector,  K,  which,  except  in  a  given  finite  region,  T, 
is  everywhere  continuous,  solenoidal,  and  lamellar,  has  in 
simply  connected  space  outside  T  an  easily  found  scalar  poten 
tial  function,  W,  which  satisfies  Laplace's  Equation.  We  may 
assign  to  W  at  pleasure  a  numerical  value  at  any  given  point, 
0,  and  define  the  value  of  W  at  any  other  point,  0',  to  be  the 
line  integral  of  the  tangential  component  of  K  taken  along 
any  path  from  0  to  0'  which  does  not  cut  T.  The  partial 
derivatives  with  respect  to  x,  y,  and  &  of  W  thus  defined 
outside  T  are  evidently  equal  at  every  point  to  the  compo 
nents  of  .A"  parallel  to  the  coordinate  axes,  and,  since  K  is 
solenoidal,  V2W=  0.  If  K  so  vanishes  at  infinity  that  the 
limit  of  the  product  of  its  intensity  and  the  square  of  the 
distance  (>•)  from  any  finite  point  is  finite,  the  limit  of 
r2  •  Dr  W  is  finite,  and  if  we  assign  to  W  the  value  zero  at 
any  point  at  infinity,  its  value  everywhere  at  infinity  will 
be  zero.  If  K  is  continuous  and  if  it  vanishes  at  infinity  in 
the  manner  just  described,  and  is  known  to  be  everywhere 
solenoidal  and  lamellar,  it  must  vanish  everywhere ;  for,  if 
we  apply  [151]  to  the  harmonic  function  W  within  an  infinite 
sphere,  it  will  appear  that  IF,  which  vanishes  at  infinity,  is 
identically  equal  to  zero.  The  vector  which  represents  the 
force  in  the  case  of  a  charged  spherical  conductor  is  solenoidal 


ELECTROSTATICS.  219 

and  lamellar  within  and  without  the  conductor,  and  it  vanishes 
properly  at  infinity,  but  it  is  discontinuous  at  the  surface  of 
the  sphere.  It  is  usually  convenient  to  assume  that  the 
integral  of  the  normal  component  of  a  vector,  taken  over  any 
closed  surface  at  which  the  vector  and  its  first  derivatives  are 
continuous,  is  equal  to  the  integral  of  the  divergence  taken 
through  the  space  within  the  surface,  even  though  at  some 
inner  surface  the  vector  is  discontinuous.  On  this  assumption 
the  vector  just  mentioned  is  not  solenoidal  on  the  surface  of 
the  conductor,  for  it  has  there  divergence  equal  in  total 
amount  to  4?r  times  the  charge. 

The  line  integral  of  the  tangential  component  of  a  vector, 
taken  around  a  closed  curve  on  which  this  component  is  con 
tinuous,  is  generally  used  as  a  measure  of  the  integral  of  the 
normal  component  of  the  curl  of  the  vector  taken  over  a  cap,  *S', 
bounded  by  the  curve,  even  though  at  some  curve  on  S  the 
vector  ceases  to  be  continuous. 

A  vector  cannot  be  considered  lamellar  at  a  surface  where, 
though  its  normal  component  is  continuous,  some  of  its  tangen 
tial  components  are  discontinuous. 

If  two  continuous  vectors,  U  and  U',  which  so  vanish  at 
infinity  that  r'2U  and  r*U'  have  finite  limits,  have  at  every 
point  in  space  equal  curls  and  divergences,  and  are  lamellar 
and  solenoidal  outside  certain  given  finite  regions,  they  are 
identical ;  for  the  difference  between  these  vectors  is  every 
where  lamellar  and  solenoidal,  and  it  vanishes  at  infinity  in 
such  a  manner  that  the  product  of  its  intensity  and  the 
square  of  the  distance  from  any  finite  point  is  finite.  This 
theorem  may  be  extended  to  the  case  where  U  and  U',  though 
not  everywhere  continuous,  have  identical  discontinuities. 

If  Nj,  &,  T/!,  £t  represent  the  numerical  values  at  the  point 
(XD  y\i  %i)  of  the  divergence  and  the  curl  components  of  the 
vector  U,  which  outside  a  given  region  is  everywhere  continu 
ous,  lamellar,  and  solenoidal.  and  which  so  vanishes  at  infinity 
that  r2  U  has  a  finite  limit ;  if 


220  ELECTROSTATICS. 

and  if 


in  which  the  integrations  are  to  he  extended  over  all  space,  or 
at  least  over  all  space  where  U  is  not  lamellar  and  solenoidal  ; 
we  know  from  the  theory  of  the  Newtonian  potential  function, 
where  similar  integrals  have  been  studied,  that,  if  N,  £,  77,  £ 
are  the  divergence  and  the  curl  components  of  U  at  (x,  y,  «), 

V*E  =  N,  V*FX  =  -  fe  ^Fy  =  -  I,,  V*FZ  =  -  f. 

The  divergence  of  the  vector  F,  which  has  the  components 
Fx,  Fy,  Fs)  is  equal  to 


and,  since  Z>.,(l/r)  =  -  DXl(l/r), 

and  -  ft  •  D,  (1  /r)  =  J)^,  /r-DXi  (^/r), 

we  may  write  this  by  the  help  of  Green's  transformation  in 
the  form 

*ti  +  D-^  +  D^}  /r'dTl 

[f  ,  •  cos  (x,  w)  +  17,  •  cos  (y,  w)  +  {i  •  cos  («,»)]  /r  •  rf/S,, 


where  the  second  integral  is  to  be  taken  over  the  outer  boundary 
of  space.  The  integrand  of  the  triple  integral  vanishes  every 
where,  because  the  vector  (£,  rj,  £),  being  the  curl  of  another 
vector,  is  itself  solenoidal.  The  field  of  the  double  integral  is 
in  a  region  where  U  is  lamellar,  so  that  the  integral  itself 
vanishes  and  F  is  seen  to  be  solenoidal  for  all  values  of  x,  y, 
and  3. 

From  these  results  it  appears  that  the  vector  which  has  for 
its  components  (DXE  plus  the  x  component  of  the  curl  of  F), 
(DyE  plus  the  y  component  of  the  curl  of  F),  (DZE  plus  the 


ELECTROSTATICS.  221 

z  component  of  the  curl  of  F)  has  everywhere  the  same  curl 
and  the  same  divergence  as  U  and  vanishes  like  it  at  infinity, 
so  that  it  is  identically  equal  to  U.  DXE,  DyE,  DZE  are 
the  components  of  a  lamellar  vector,  and  the  curl  of  F  is 
solenoidal,  so  that  the  vector  U,  which  is  not  everywhere 
either  solenoidal  or  lamellar,  is  everywhere  expressible,  as 
was  first  shown  by  Helmholtz,*  as  the  sum  of  a  solenoidal 
and  a  lamellar  vector.  The  equations 

Ux  =  DXE  +  DyFs  -  DzFy,        Uy  =  DyE  +  DZFX  -  DXFZ, 


give  any  vector,  U9  which  is  known  to  vanish  properly  at 
infinity,  when  its  curl  components  and  its  divergence  are 
known.  If  U  is  solenoidal,  E  vanishes  and  F  is  a  vector 
potential  function  of  U.  Every  lamellar  vector  has  a  scalar 
potential  function  the  component  of  the  gradient  of  which,  at 
any  point,  in  any  direction,  gives  the  intensity  of  the  compo 
nent  of  the  vector  at  that  point,  in  that  direction.  The  com 
ponent  at  any  point,  in  any  direction,  of  the  curl  of  a  vector 
potential  function  of  a  solenoidal  vector  gives  the  intensity  of 
the  component  of  the  vector  at  the  given  point,  in  the  given 
direction.  Heaviside  gives  the  name  "circuital"  to  a  vector 
which  is  solenoidal  but  not  lamellar,  and  the  name  "diver 
gent  "  to  a  vector  which  is  lamellar  but  not  solenoidal. 

If  pl  is  a  function  of  xlf  y^  «1?  and  if  r2  stands  for  the 
expression  (x  -  a^)2  +  (y  —  ytf  -f  (z  —  z^2,  the  familiar  inte 

gral    ill  —  dx^du^dz^  extended  over  all  space,  is  a  function 

of  X,  //,  Zj  which  Prof.  J.  Willard  Gibbs  in  a  remarkable  paper  t 
has  denoted  by  the  symbol  Pot  p.  Using  this  notation,  we 
may  write 

47r£'=-PotN,  47r^  =  Pot£,  4  vFy  =  Pot  77,  4  ir^  =  Pot  £  ; 


*  Crelle's  Journal.  Bd.  LV,  1858. 

t  Elements  of  Vector  Analysis,  §  92.     See  also  Heaviside's  Electrical 
Papers,  XXIV. 


222  ELECTKOK1NEMATICS. 

and  if  we  represent  by  Pot  curl  U  the  vector  which  has  for 
its  components  Pot  £,  Pot  77,  Pot  £,  we  have  the  vector  equation 
4  irF=  Pot  curl  V,  and  if  U  is  solenoidal,  4  TT  U=  curl  Pot  curl  U. 
If  U  is  solenoidal,  4  TT  £7  —  curl  curl  Pot  CT  =  Pot  curl  curl  U, 
and  curl  Pot  U  is  a  vector  potential  function  of  kirU,  or 
Pot  £7  is  a  vector  potential  function  of  a  vector  potential 
function  of  ±irU.  In  the  case  of  any  polarized  distribution 
whatever,  provided  there  is  no  intrinsic  volume  density  p0,  the 
induction  is  solenoidal  and  has  a  vector  potential  function. 


II.    ELECTROKINEMATICS. 

70.  Steady  Currents  of  Electricity.  When  a  charged  body 
A  is  brought  up  into  the  neighborhood  of  a  previously 
uncharged,  insulated  conductor  J5,  the  two  kinds  of  elec 
tricity  which,  according  to  our  provisional  theory,  exist  in 
equal  quantities  in  every  particle  of  B  tend  to  separate 
from  each  other  and,  as  a  consequence,  free  electricity 
appears  on  B's  surface,  some  parts  of  this  surface  becoming 
charged  positively  and  other  parts  negatively.  If  A  is  brought 
into  a  given  position  and  fixed  there,  the  distribution  on  the 
surface  of  B  quickly  attains  and  keeps  a  value  determined  by 
the  fact  that  the  whole  interior  of  B  must  be  a  region  at  con 
stant  potential,  or,  in  other  words,  that  the  resultant  force  at 
any  point  within  B  due  to  the  free  electricities  on  its  surface 
must  be  equal  and  opposite  to  the  force  at  that  point  due  to  all 
the  free  electricity  outside  B.  If,  now,  A  with  its  charge  is 
moved  to  a  new  position,  the  old  distribution  on  2?'s  surface  will 
not  in  general  screen  the  interior  of  B  from  the  action  of  ^4's 
charge,  and  a  new  separation  of  electricity  within  B  and  a  new 
arrangement  or  combination  of  the  charge  on  the  surface  is 
necessary  before  a  new  state  of  equilibrium  can  be  established. 
If  A  be  moved  continuously  in  any  manner,  there  will  be  a  con 
stant  attempt  on  the  part  of  the  separated  electricities  to  set 


ELECTROKIXEM  AT  LCS.  -  -  8 

up  a  state  of  equilibrium,  and  hence  at  every  point  of  B  there 
will  be,  in  general,  some  electrical  change  going  on  continually. 

If  two  conductors  ^1  and  B  at  different  potentials  be  con 
nected  by  a  fine  Avire,  the  whole  will  form  a  single  conductor, 
which  can  only  be  in  a  state  of  equilibrium  when  the  value  of 
the  potential  function  due  to  all  the  free  electricity  in  existence 
is  constant  throughout  its  interior,  and  there  will  be  such  a 
transfer  of  electricity  through  the  wire  as  will  establish  this 
state  of  equilibrium  in  a  very  short  time.  If,  however,  by  any 
device  we  can  furnish  unlimited  quantities  of  electricity  to  A 
and  B  in  such  a  way  as  to  keep  them  at  the  same  potentials  as 
at  the  beginning,  there  will  be  a  continual  attempt  to  establish 
electric  equilibrium  within  the  compound  conductor  consisting 
of  A,  B,  and  the  wire,  and.  as  a  result,  there  will  be  a  continual 
transfer  of  electricity  through  the  wire. 

The  transfer  of  electricity  from  one  place  to  another  through 
a  conductor  is  a  very  common  phenomenon.  Sometimes,  as  we 
have  seen,  electricity  traverses  the  conductor  for  a  short  time 
only ;  sometimes,  however,  the  transfer  goes  on  indefinitely, 
and,  so  far  as  we  can  judge  from  its  attendant  phenomena,  at 
a  constant  rate,  so  that  just  as  much  of  a  given  kind  of  elec 
tricity  crosses  any  surface  within  the  conductor  in  any  one 
second  as  in  any  other :  such  a  continuous  steady  transfer  as 
this  is  called  a  "  steady  current." 

The  existence  of  a  steady  current  in  a  conductor  implies  a 
force  tending  to  drive  electricity  through  the  conductor  ;  that  is, 
it  implies,  at  least  in  the  absence  of  moving  magnetic  masses 
and  of  electric  currents  in  the  neighborhood  of  the  conductor, 
free  electricity  somewhere  in  existence  which  gives  rise  to  a 
potential  function  not  constant  throughout  the  conductor.  No 
pail  of  a  conductor  through  which  a  steady  current  is  flowing- 
can  accumulate  free  electricity  as  the  time  goes  on,  for  such  an 
accumulation  increasing  with  the  time  would  be  accompanied 
by  changes  which  must  show  themselves  outside  the  conductor. 
We  are  led  to  assume,  then,  that  if  any  closed  surface  be  drawn 
inside  a  conductor  which  carries  a  steady  current,  just  as  much 


224  ELECTROKINEMATICS. 

electricity  of  a  given  kind  enters  the  region  enclosed  by  the 
surface  in  any  interval  of  time  as  leaves  it  during  that  interval. 

We  have  seen  that  at  every  point  inside  a  conductor  where 
there  is  a  resultant  electric  force  there  will  be  an  electric  sepa 
ration  which  will  go  on  as  long  as  the  force  exists.  Experi 
ment  seems  to  show  that  the  rate  of  separation  of  quantities  of 
electricity  is  proportional  to  the  magnitude  of  the  force.  Let 
P  be  a  point  of  a  small  plane  area  u>  inside  a  conductor,  and 
let  F  be  the  average  value  during  the  interval  from  t  to  t  -f-  A£ 
of  the  component  of  the  electric  force  normal  to  this  area ;  then 
in  what  follows  we  shall  assume  that  the  amount  of  positive 
electricity  which  crosses  this  surface,  in  the  sense  in  which  the 
force  points,  during  the  interval  is  k  •  w  •  F*  A£,  where  A;  is  a  con 
stant  depending  only  upon  the  material  of  which  the  conductor 
is  composed  and  upon  its  physical  condition.  The  average 
value  of  this  flux  per  unit  of  time  per  unit  of  surface  is,  there 
fore,  k  •  F.  If,  now,  <o  and  A£  are  made  to  grow  smaller  and 
smaller  in  such  a  manner  that  P  is  always  a  point  of  o>,  F  ap 
proaches  as  a  limit  the  negative  of  the  value  at  P  of  the  deriva 
tive,  taken  in  the  direction  in  which  F  acts,  of  F,  the  potential 
function  due  to  all  the  free  electricity  in  existence  ;  so  that  at 
any  instant  the  value  at  a  point,  P,  in  any  direction,  n,  of  the 
rate  of  flow  of  positive  electricity  across  a  surface  normal  to  ??, 
per  unit  of  this  surface  per  unit  of  time,  is  the  value  at  P 
of  -k-DnV. 

It  follows  from  this  that  if  any  tube  of  force  be  drawn  in  a 
conductor  which  carries  a  steady  current,  there  is  no  flow 
through  the  sides  of  the  tube.  Consider  a  region  shut  in  by  a 
tube  of  force  and  by  two  equipotential  surfaces  inside  a  con 
ductor  through  which  a  steady  current  is  flowing.  Let  o^  and  w2 
be  the  areas  of  the  equipotential  ends  of  the  region,  and  let  Fl 
and  F2  be  the  average  values  of  the  normal  force,  taken  in  the 
same  sense  in  both  cases,  over  these  ends.  Applying  Gauss's 
theorem  to  this  region  we  have  F2w2  —  F^  =  47rQ,  where  Q  is 
the  amount  of  free  electricity,  algebraically  considered,  within 
the  region.  If  the  conductor  is  homogeneous,  the  amount  of 


ELECTROKINEMATICS.  225 

positive  electricity  which  enters  —  or  the  amount  of  negative 
electricity  which  leaves  —  the  region  by  one  end  per  unit  of 
time  is  kFl  •  o^,  and  the  amount  which  leaves  it  at  the  other  end 
is  kF2 •  tu2.  These  amounts  are  equal,  so  that  jF2w2  —  Flw1  =  0  ; 
hence,  Q  =  0,  and  there  is  no  free  electricity  at  any  point  within 
a  homogeneous  conductor  which  carries  a  steady  current.  The 
free  electricity  which  gives  rise  to  the  potential  function  the 
rate  of  change  of  which  is  proportional  to  the  flow  of  electricity 
within  the  conductor,  must  then  lie  either  outside  the  conduc 
tor,  or  on  its  surface,  or  both.  It  would  not  be  difficult  to 
prove  that  there  must  be  a  distribution  of  electricity  on  parts 
of  the  surface  of  even-  conductor  which  carries  a  steady  current 
and  is  in  contact  in  some  places  with  an  insulating  medium  ; 
but  the  fact  that  a  wire  through  which  such  a  current  is  passing 
may  be  moved  about  so  as  to  change  its  position  with  respect 
to  outside  bodies  without  changing  the  amount  of  the  current 
will  suffice  to  make  it  probable  that  a  part,  at  least,  of  the  free 
electricity  that  we  have  been  considering  moves  with  the  wire. 
Since  the  density  of  the  free  electricity  within  a  conductor 
which  carries  a  steady  current  is  zero,  the  potential  function 
F,  inside  the  conductor,  must  satisfy  Laplace's  Equation ; 
that  is,  V2F=0.  It  is  easy  to  see,  since  there  can  be  no 
accumulation  of  free  electricity  in  any  conductor  which  bears 
a  steady  current,  that  the  amount  of  electricity  which  comes 
up  on  one  side  to  the  common  surface  of  two  such  conductors 
which  are  in  contact  must  be  equal  to  that  which  goes  away 
from  this  surface  on  the  other ;  that  is,  at  every  point  of 
the  surface,  l\  •  DnVi  =  A*2  •  Z).,F2,  where  k\  and  A\>  are  the  spe 
cific  conductivities  of  the  two  conductors,  and  DnVl  and  Dn F2 
the  values  at  the  point,  taken  in  the  same  sense  in  both  cases, 
of  the  derivatives  of  F  in  the  direction  of  the  normal  to  the  sur 
face,  one  on  one  side  of  the  surface,  and  the  other  on  the  other. 
It  is  to  be  noticed  that  the  boundary  between  two  such  con 
ductors  may  or  may  not  be  an  equipotential  surface.  At  every 
point  of  the  common  surface  of  a  conductor  and  an  insulating 
medium  k-DnV=Q  or  D,,F=0;  hence  the  equipotential  sur- 


226  ELECTROKINEMAT1CK. 

faces  within  the  conductor  cut  the  surface  where  the  conductor 
abuts  on  the  insulating  medium  at  right  angles. 

70.   Linear  Conductors.    Resistance.    Law  of  Tensions.     Let 

us  consider  the  case  of  a  linear  conductor,  that  is,  one  in  which 
all  the  lines  of  force  are  parallel  to  each  other  and  to  the  sides 
of  the  conductor,  so  that  every  tube  of  force  has  a  constant 
cross-section  throughout  that  part  of  its  length  which  lies  in  the 
given  conductor.  It  will  appear  later  on  that  any  right  cylin 
drical  conductor,  whatever  the  form  of  its  cross-section,  will  be 
a  linear  conductor,  if  every  point  of  one  of  its  ends  be  kept 
at  one  constant  potential,  and  every  point  of  the  other  end  at 
another.  It  will  also  be  evident  that  such  wires  as  are  ordinarily 
used  for  making  electrical  connections  are,  to  all  intents  and 
purposes,  except  perhaps  at  the  very  ends,  linear  conductors, 
whether  these  wires  are  straight  or  curved.  Let  the  ends  of  a 
homogeneous  long  uniform  straight  wire  of  constant  cross- 
section  q,  and  of  length  /,  be  kept  respectively  at  potentials 
V  and  F".  Take  the  axis  of  the  wire  for  the  axis  of  aj,  and 
the  origin  at  that  end  of  the  wire  at  which  the  potential  func 
tion  due  to  all  the  free  electricity  in  existence  is  F'  ;  then  every 
line  of  force  inside  the  wire  is  parallel  to  the  axis  of  x  ;  and 
since  there  is  no  force  in  any  direction  perpendicular  to  the 
axis  of  a?,  Z>3/F=01,  Z)2F=0,  and  Laplace's  Equation,  which 
must  be  satisfied  by  F  inside  the  wire  becomes  Z)/F=0, 
whence  V—  Ax  -f-  B  ;  or,  since  V—  V  when  x  =  0,  and  F=  F" 
when  x  =  l, 

r=(V»-V')X     |      y, 

The  steady  current  c  which  traverses  the  wire  carries  across 
every  right  section  in  the  unit  of  time  —  kg  •  DXV  units  of  posi 
tive  electricity  in  the  positive  direction  of  the  axis  of  x.  That  is, 


where  k  is  the  specific  conductivity  of  the  material  out  of  which 


ELECTROKINEMATICS.  'I'l  i 

the  wire  is  made.  The  quantity  l/kq  is  called  the  resistance 
of  the  wire,  kq/l  its  conductivity.  The  quantity  &  is  a 
function  of  the  temperature.  In  the  case  of  a  pure  solid 
metal  at  any  ordinary  temperature  a  rise  of  1°  C.  will 
increase  1/k  by  about  0.004  times  its  own  value.  This 
fractional  increase  is  much  smaller  in  the  case  of  some 
alloys :  for  "  manganin  "  at  room  temperatures  it  is  not  more 
than  0.00001. 

The  analysis  of  this  section  assumes  that  the  homogeneous 
linear  conductor  is  at  the  same  temperature  throughout  and 
that  it  is  not  surrounded  by  a  changing  magnetic  field. 

It  is  an  important  physical  principle,  first  enunciated  in  a 
slightly  different  form  by  Ohm,  that  if  a  fixed  portion  of  the 
surface  of  a  given  homogeneous  conductor  be  kept  constantly 
at  potential  V^  and  another  fixed  portion  at  potential  VZ)  while 
the  rest  of  the  surface  of  the  conductor  is  in  contact  with  an 
insulating  medium,  the  ratio  of  V±  —  V»  to  the  steady  current 
which  traverses  the  conductor,  —  as  measured  by  the  quantity 
of  positive  electricity  per  unit  of  time  which  either  enters  the 
conductor  through  the  surface  V  =  J\  or  leaves  it  through  the 
surface  V  =  F2,  —  is  a  quantity  independent  of  Vl  and  F2. 
This  ratio  is  called  the  resistance  of  the  conductor  under  the 
given  circumstances.  The  resistance  of  a  conductor  depends 
not  only  upon  its  shape,  the  material  of  which  it  is  composed, 
and  the  temperature  and  other  physical  conditions  of  this 
material,  but  also  upon  the  shape,  size,  and  position  of  those 
portions  of  the  surface  which  are  kept  at  the  potentials  V\  and 
F2.  The  resistance  of  so  much  of  a  tube  of  force  drawn  in  a 
conductor  which  bears  a  steady  current  as  lies  between  the 
equipotential  surfaces  V  —  J\  and  V  =  Vz  is  the  ratio  of  V\  —  Vz 
to  the  amount  of  positive  electricity  per  unit  of  time  which 
enters  the  portion  of  the  tube  which  we  have  been  considering 
through  the  surface  V  =•  Fi,  or  leaves  it  through  the  surface 
V=  V2,  or  crosses  any  section  of  the  tube  in  the  direction  indi 
cated.  Any  electric  change  which,  under  the  same  conditions 
of  temperature  and  pressure,  will  leave  this  tube  of  force  still 


228  ELECTROKINEMATICS. 

a  tube  of  force  and  its  equipotential  ends  still  equipotential, 
however  the  value  of  the  potential  function  may  be  changed, 
will,  according  to  this  law  of  Ohm,  leave  the  resistance  the 
same.  Other  things  being  equal,  the  resistance  of  a  tube  of 
force  increases  with  the  length  of  the  tube  and  diminishes  as 
the  section  of  the  tube  is  made  greater. 

Suppose  that  we  have  a  series  of  linear  conductors  joined 
end  to  end  in  a  closed  ring,  so  that  the  end  of  the  nth  conductor 
is  in  contact  with  the  beginning  of  the  first.  Let  Fm'  and  Vm" 
be  the  values  of  the  potential  function  at  the  beginning  and  end 
of  the  rath  conductor,  and  rm  the  resistance  of  this  conductor. 
Since  the  same  current  c  must  traverse  every  conductor  of  the 
series,  we  have 

77' 77  it — „„      77' 77  f — »„     77' 77" — „„  77' 77" — »y.  . 

'1  '1     -~crl)     r 2  '2     — 672>     '3  '3     ~ '  Lr  3?    ''''n  Vn     ~~6/M> 

and,  if  we  add  them  together,  we  shall  get 


rl  +  r2  +  rs-\ -f  rn 

where  F2'  —  FI"  is  the  difference  between  the  values  of  the 
potential  function  on  opposite  sides  of  the  surface  common  to 
the  second  and  first  conductors,  F3'  —  F2"  the  corresponding 
difference  for  the  third  and  second  conductors,  and  so  on 
around  the  ring.  If  these  differences  are  not  all  zero,  the 
circuit  is  said  to  be  the  seat  of  an  electromotive  force. 

We  may  here  assume  that  when  any  two  conductors,  at 
the  same  temperature  throughout,  but  made  of  different  mate 
rials,  are  placed  in  contact  with  each  other,  a  discontinuity* 
of  the  potential  function  suddenly  appears  at  their  common 

*  Although  the  language  of  the  old  "Two  Fluid  Theory"  is  used  in 
this  chapter,  the  reader  is  strongly  urged  to  make  himself  acquainted  with 
the  physical  theories  now  commonly  used  in  accounting  for  electrical 
phenomena.  See  Dr.  0.  J.  Lodge's  papers  "  On  the  Seat  of  the  Electro 
motive  Force  in  the  Voltaic  Cell,"  printed  in  the  Philosophical  Magazine 
for  March,  April,  May,  and  October  of  1885,  and  his  "  Modern  Views  of 
Electricity,"  a  series  of  contributions  to  Nature,  begun  in  1886. 


ELECTROKINEMATICS.  229 

surface.  The  amount  of  this  discontinuity,  which  remains  con 
stant  after  it  has  once  been  established,  is  the  same  for  all 
points  of  the  common  boundary  of  the  two  conductors,  and  is 
independent  of  their  size  and  shape,  of  the  extent  of  surface  in 
contact,  and  of  the  absolute  values  of  the  potential  function  on 
either  side  of  the  boundary.  We  shall  represent  the  sudden 
fall  in  the  value  of  the  potential  function  encountered  by  pass 
ing  from  a  conductor  made  of  material  A  to  a  conductor  made 
of  material  B  across  any  point  of  their  common  surface  b}~  the 
symbol  A  \  B.  A  certain  class  of  substances,  to  which  all 
metals  belong,  has  the  property  that  if  L,  M,  and  y  are  any 
three  of  these  substances,  all  at  the  same  temperature, 

L  |  J/+J/|  X=L  |  JV. 

This  class  is  said  to  obey  "  Volta's  Law  of  Tensions."  If  a 
number  of  conductors  made  of  different  kinds  of  metals  all  at 
the  same  temperature  be  placed  in  line,  the  first  in  contact  with 
the  second,  the  second  with  the  third,  and  so  on,  the  algebraic 
sum  of  the  jumps  of  the  potential  function  encountered  in  going 
from  the  first  conductor  to  the  last  through  all  the  others  is 
exactly  the  same  in  amount  as  the  single  jump  which  would 
occur  at  the  common  surface  of  the  first  and  last  conductors  if 
they  were  put  directly  in  contact  with  each  other.  Some  other 
substances  besides  metals  obey  the  Law  of  Tensions,  but  most 
liquids  and  solutions,  whether  in  contact  with  each  other  or  with 
metals,  do  not  obey  this  law. 

The  sum  of  the  jumps  in  the  potential  function  encountered 
in  passing  from  copper  to  zinc  by  way  of  an  iron  conductor  is 
the  same,  if  the  whole  be  at  one  temperature,  as  the  jump 
encountered  in  passing  directly  from  copper  to  zinc.  But  this  is 
not  equal  to  the  sum  of  the  jumps  met  with  in  passing  from 
copper  to  zinc  through  sulphuric  acid. 

Cu  |  Fe  +  Fe  |  Zn=Cu  |  Zn, 
but  Cu  |  (H2SO4)  +  (H2SO4)  |  Zn  ^  Cu  |  Zn. 

The  numerator  of  the  expression  just  found  for  the  intensity 
of  the  current  which  traverses  a  closed  chain  of  linear  conduc- 


230  ELECTKOKLNKALATICS. 

tors  is  evidently  the  algebraic  sum  of  the  "jumps"  in  the 
potential  function  encountered  by  travelling  in  the  direction 
in  which  the  current  is  supposed  to  move,  from  the  first 
conductor  to  the  last  through  all  the  others,  and  reckoning 
the  jump  at  any  boundary  positive  if  the  value  of  the  poten 
tial  function  is  increased  as  one  crosses  the  boundary.  If  all 
the  conductors  which  form  the  circuit  are  metallic  and  all  at 
the  same  temperature,  whether  or  not  they  are  all  made  of  the 
same  kind  of  metal,  this  numerator  is  zero,  and  it  follows  that 
in  order  that  a  steady  current  may  traverse  a  circuit  of  con 
ductors,  one  at  least  of  the  conductors  must  disobey  the  Law 
of  Tensions. 

The  same  formulas  apply  to  a  circuit  composed  of  conduc 
tors  of  any  form  if  each  of  the  common  surfaces  of  contigu 
ous  conductors  is  equipotential. 

Every  slender  tube  of  force  in  a  homogeneous  conductor 
which  carries  a  steady  current  is  also  a  tube  of  flow  and 
constitutes  a  current  filament.  We  shall  hereafter  apply 
the  term  linear  only  to  conductors  which  have  very  small 
cross-sections. 

72.  Electromotive  Force.  We  have  seen  that  if  a  number 
of  homogeneous  conductors  made  of  different  materials  be 
connected  in  series  to  form  a  heterogeneous  conductor  K, 
there  will  be  discontinuities  in  the  electrostatic  potential 
function  within  K  at  the  common  surfaces  of  adjacent  con 
ductors.  If  an  equipotential  surface  A  near  one  end  of  K  be 
kept  at  potential  VA,  and  an  equipotential  surface  B  near  the 
other  end  of  Ky  at  potential  VB,  and  if  the  algebraic  sum  of 
the  discontinuities  of  potential  between  A  and  B,  counting  a 
step  up  as  positive,  is  E,  the  current  in  K  from  A  to  B  will  be 
(  VA  —  VB  -f-  E)  /r,  where  r  is  the  resistance  between  A  and  />. 
In  such  a  case  as  this,  VA  —  VR  is  called  the  electrostatic  or 
external  electromotive  force,  and  E  the  internal  or  intrinsic  elec 
tromotive  force.  If  K  forms  a  closed  circuit,  all  the  electro 
motive  force  may  be  regarded  as  internal.  In  this  connection 


ELECTROKINEMATICS. 


231 


it  should  be  said  that,  although  physicists  are  not  all  in 
agreement  as  to  the  magnitude  of  the  discontinuity  of  poten 
tial  at  the  surface  of  contact  of  any  two  given  dissimilar 
conductors,  there  is  no  difference  of  opinion  as  to  the  alge 
braic  sum  of  these  discontinuities  in  the  case  of  any  closed 
circuit. 

If  one  end  of  a  hetero 
geneous  cylindrical  con 
ductor  K,  of  given 
resistance  r,  formed  of 
homogeneous  cylindrical 
conductors  in  series,  be 
kept  at  a  given  poten 
tial  F!  and  the  other  end 
at  the  given  potential  F2, 
the  value  of  the  potential 
function  will  depend  very 
much  upon  the  constitu 
tion  of  K.  Three  different 
cases  are  illustrated  in 
Fig.  56,  in  which  abscis 
sas  represent  resistances 
and  ordinates  the  corre 
sponding  values  of  V. 
In  these  figures  A  is  sup 
posed  to  be  an  electro 
lyte,  while  Z,  M,  N  are 
metals  :  Fx  =  2,  F2  =  0.5, 
A\N=O.S,  A\M=1.S, 
N\M=Q.  The  current 

strength  (indicated  by  the  slope  of  the  line  which  gives 
the  value  of  F)  is  evidently  different  in  the  different 
diagrams. 

Fig.  57  represents  V  in  a  long  chain  made  of  two  metals  P, 
Q,  and  an  electrolyte  7?,  such  that  R  \  P  is  small,  R  \  Q  still 
smaller,  and  P  \  Q  zero.  Here  the  ends  are  at  the  same 


FIG.  56. 


232 


ELECTKOKINEMATICS. 


potential,  and  there  are  no  great  potential  differences  any 
where  in  the  chain,  but  the  current  (as  indicated  by  the 
slope  of  the  F  line)  is  large,  as  is  the  sum  of  the  small 
discontinuities  which  go  to  make  up  the  electromotive  force 
in  the  chain. 

A  galvanic  battery  may  be  regarded  as  a  chain  of  three 
or  more  generally  non-linear  conductors,  at  least  one  of  which 
disobeys  the  Law  of  Tensions.  The  algebraic  sum  of  the 
jumps  in  the  potential  function  encountered  by  starting  at 
that  pole  of  a  galvanic  battery  at  which  the 
potential  is  less,  and  passing  to  the  other 
pole  through  the  battery,  is  the  electromotive 
force  of  the  battery.  The  difference  of  poten 
tial  between  the  poles  of  the  battery,  when  they 
are  not  connected,  measures  this  electromotive 
force.  Chemical  action  goes  on  inside  every 
FIG.  57.  battery  when  its  poles  are  closed ;  some  of  its 
solutions  are  decomposed,  and  the  products  of 
this  decomposition  often  appear  at  the  boundaries  of  the  liquid 
conductors  inside  the  battery  and  decrease  the  electromotive 
force  by  changing  the  amount  of  jump  in  the  potential  func 
tion  at  each  of  these  boundaries.  For  this  reason  the  electro 
motive  force  of  a  battery  in  action  may  be  much  less  than 
when  the  poles  are  open. 

If  two  points,  P  and  Q,  in  a  network  of  conductors  which 
carry  a  steady  current,  be  connected  by  an  additional  wire 
conductor,  K,  containing  a  battery  of  such  electromotive  force, 
e,  and  so  directed  as  to  prevent  any  current  from  passing 
through  K,  e  measures  the  difference  of  potential  between  P 
and  Q.  It  is  easy  to  show  that  when  the  poles  of  a  battery 
are  closed  by  a  conductor  of  resistance  E,  the  difference 
between  the  values  of  the  potential  function  at  the  ends  of 
this  conductor  is  RE  /  (B  +  R),  where  E  is  the  electromotive 
force  of  the  battery  under  the  given  circumstances,  and  B 
the  resistance  of  the  conductors  which  make  up  the  battery 
itself.  The  steady  current  which  flows  through  the  circuit 


ELECTROKINEMATICS.  233 

carries  E  /  (B  -f-  7?)  units  of  positive  electricity  across  every 
cross-section  per  unit  of  time.  With  a  given  battery  the 
intensity  of  the  current  can  be  changed  very  much,  of  course, 
by  increasing  or  decreasing  the  resistance  of  that  part  of  the 
circuit  which  lies  outside  the  battery. 

In  the  centimetre-gramme-second  system  of  electrostatic 
[E.S.]  absolute  units,  the  unit  of  electric  quantity  is  that 
quantity  of  electricity  which,  if  it  could  be  concentrated 
at  a  point  in  air,  would  repel  a  like  quantity  concentrated  at 
a  point  1  centimetre  from  the  first  with  a  force  of  1  dyne. 
This  unit  is  found  inconveniently  small,  however,  when  one 
has  to  deal  with  such  steady  currents  as  are  usually  met 
with  in  practice,  and  the  coulomb,  which  is  equal  to  about 
3  x  10°  of  these  absolute  units,  is  the  practical  unit  of 
quantity  most  frequently  used. 

The  absolute  E.S.  unit  of  current  carries  the  absolute  unit 
of  electricity  past  any  point  in  its  course  each  second.  A 
current  of  a  coulomb  per  second  (equivalent  to  3  X  109  of 
these  absolute  current  units)  is  called  an  ampere. 

The  absolute  E.S.  unit  of  resistance  is  9  X  1011  times  as 
large  as  the  practical  unit  called  the  ohm.  The  latter  is  the 
resistance  of  a  column  of  pure  mercury  1  square  millimetre 
in  section  and  106.3  centimetres  long,  at  0°  C.  The  resist 
ance  at  0°  C.  of  a  wire  of  pure  copper  1  millimetre  in  diameter 
and  1  metre  long  is  about  0.01642  ohm. 

The  absolute  E.S.  unit  of  difference  of  potential  is 
equivalent  to  300  practical  units.  The  practical  unit, 
called  the  volt,  is  such  that  if  the  two  ends  of  a  wire  of 
1  ohm  resistance  were  kept  at  1  volt  difference  of  poten 
tial,  the  steady  current  which  traversed  the  wire  would 
carry  past  any  cross-section  1  coulomb  of  electricity  per 
second. 

A  condenser  which  requires  1  coulomb  of  electricity  to 
charge  it,  so  that  the  difference  of  potential  between  its 
poles  is  1  volt,  is  said  to  have  a  capacity  of  1  farad.  A  con 
venient  unit  of  capacity  is  the  microfarad  or  the  millionth 


234  ELECTROKINEMATICS. 

of  a  farad.  It  is  equivalent  to  900,000  absolute  E.S.  units 
of  capacity.  The  capacity  of  a  conducting  sphere  9  kilo 
metres  in  radius  would  be  1  microfarad,  that  of  the  earth 
something  over  700  microfarads.  The  capacity  of  a  nautical 
mile  of  such  ocean  telegraph  cable  as  is  usually  laid  may  be 
taken  to  be  about  4,  microfarad. 

73.  KirchhofJ's  Laws.  The  Law  of  Divided  Circuits.  From 
what  has  been  proved  in  the  preceding  sections  about  conduc 
tors  which  carry  steady  currents,  follow  two  theorems  of  much 
practical  importance,  called  KirchhofPs  Laws. 

I.  If  several  wires  which  form  part  of  a  network  of  conductors 
carrying  a  steady  current  meet  at  a  point,  the  sum  of  the  inten 
sities  of  all  the  currents  which  flow  towards  the  point  through 
these  wires  is  equal  to  the  sum  of  all  those  which  recede  from 
it ;  or,  in  other  words,  the  algebraic  sum  of  all  the  currents 
which  approach  the  point  through  the  wires  which  meet  there 
is  zero. 

II.  If,  out  of  any  network  of  wires  which  form  a  complex 
conductor  and  carry  a  steady  current,  a  number  of  wires  which 
form  a  closed  figure  be  chosen,  and  if,  starting  at  any  point, 
we  follow  the  figure  around  in  either  direction,  calling  all  cur 
rents  which  move  with  us  positive,  and  all  discontinuities  of 
the  potential  function  which  lift  us   from  places   of  lower 
potential  to  places  of  higher  potential  positive,  the  algebraic 
sum  of  the  products  formed  by  multiplying  the  resistance  of 
each  conductor  by  the  current  running  through  it,  is  equal  to 
the  algebraic  sum  of  the   jumps   in  the  potential  function 
which  we  encounter  in  going  completely  around  the  figure. 

The  first  of  these  laws  is  an  immediate  consequence  of  the 
fact  that  there  can  be  no  growing  accumulation  of  free  elec 
tricity  anywhere  in  a  circuit  which  bears  a  steady  current. 
To  prove  the  second  law,  let  a1?  a2,  as,  •  •  •  an  be  n  linear  con 
ductors,  which,  taken  in  order,  form  a  closed  figure,  itself  a 
part  of  a  complex  conductor  which  carries  a  steady  current. 
In  passing  from  a^  to  an  through  all  the  other  conductors,  let 


ELECTROKINEMATICS. 


235 


Vj  and  V"  be  the  values  of  the  potential  function  at  the 
beginning  and  end  of  the  jih  conductor,  and  let  r,.  and  cj  be 
respectively  the  resistance  of  this  conductor  and  the  value  of 
the  current  running  through  it.  Then,  from  the  definition  of 
the  term  "  resistance,"  we  have  the  following  equations  : 


or,  adding  them  all  together, 


••••-T  cHrn 

-rj-rf.+vj-vj'+ 

which  is  the  statement  of  this 
law. 

If  electricity  is  free  to  pass 
from  a  point  P  to  another  point 
P'  by  two  wires  of  resistance  t\ 
and  n,  respectively,  and  if  a 
steady  current  be  flowing  from 
P  to  P',  the  current  will  be 
divided  between  the  two  wires 
in  the  inverse  ratio  of  their 
resistances  or  in  the  direct  ratio  FIG.  58. 

of  their  conductivities.     For,  if 

V  and  V  be  the  values  of  the  potential  function  at  P 
we  have  V—  V  =  c^  and  V—  V  =  c.2r.2,  whence  ^  :  cn 


and  P', 
=  r,  :  /v 


Moreover, 


or 


Cl  +  <•„ 

v-  r 


-  (  V  -  V) 


-1-  +  -  )  ; 
>-i      rj 


The  expression  in  the  second  number  of  the  last  equation 
is,  by  the  definition  of  the  term,  the  resistance  of  the  com 
pound  conductor  formed  of  the  two  which  join  P  and  P'. 
It  is  evident  that  the  conductivity  of  this  conductor  is  the 


236  BLECTROKINEMATICS. 

sum  of  the  conductivities  of  the  two  wires  of  which  it  is 
composed. 

If  n  conductors  be  joined  up  in  parallel  to  form  a  compound 
conductor,  the  conductivity  of  the  latter  is  the  sum  of  the 
conductivities  of  the  constituents,  and  its  resistance  is  the 
reciprocal  of  the  sum  of  the  reciprocals  of  their  resistance. 

If  four  conductors  the  resistances  of  which  are  p,  q,  r,  and 
s  form  a  quadrilateral  (Fig.  58)  one  pair  of  vertices  of  which 
are  connected  by  a  wire  of  resistance  g  and  the  other  pair  by  a 
conductor  of  resistance  b  containing  a  battery  of  electromotive 
force  E,  we  have  an  arrangement  of  much  practical  importance, 
which  is  often  called  Wheatstone's  Net.  If  we  denote  the 
strength  of  the  current  through  the  cell,  in  the  direction  indi 
cated  by  the  arrow  in  the  figure,  by  (7,  and  the  currents  in 
the  other  conductors  by  Cp,  CQ,  Cr,  Cs,  and  Cg  respectively, 
Kirchhoff's  Laws  yield  the  equations 

c  =  cp  +  cq  =  cr+c,,  cp  =  cg  +  cr, 

Cq=Cs-Cg,  p.Cp-q.Cq  +  ff.Cg  =  0, 


If  we  substitute  the  values  of  C,  Cp,  Cq  obtained  from  the  first 
three  equations  in  the  last  three,  we  shall  get  a  system  of 
three  linear  equations  involving  the  three  unknown  quantities 
Cg,  Cr,  Ca,  which  can  be  easily  solved.  These  equations  are 


and  if  we  denote  the  determinant  of  the  coefficients, 

-  \  pr(q  +  *)  +  qs  (p  +  r)  +  b  (p  +  q)  (r  +  s) 
+  g[b(p  +  q  +  r  +  s)  +  (q  +  s)(p  + 
or  -gr(b  +  q  +  8)  +  qb(g  +  r  +  s)  —  ps2  +  qrs 


ELECTROKIXEMATICS.  237 

it  is  easy  to  see  that 

Cg  =  E(qr-ps)/\ 

Cr  =  E(gq-\-  $2)  +  sg  4-  sq)  /  A, 


Cq  =  E  (g  r  +  gp  +  rp  +  ps)  /  A  . 

C7  =  E  (gq  -\-  sp  +  sg  -f  sq  +  pg  +  rp  +  rg  +  rq)  /  A. 

The  resistance  (It)  of  the  net  pqrsg,  computed  from  the  equa 
tion  C—E/(b  +  R),  is 

[g  (g  +  s)  (p  +  r)  +  j?r  (y  +  s)  +  qs  (p  +  r)] 

[<7  O  +  ?  +  >'  +  s}  +  (p  +  q)  (r  +  *)] 

If  no  current  passes  through  the  resistance  g,  we  have 
qr  =ps,  Cp  =  Cr,  Cq  =  Ca  and,  as  we  may  see  by  multiplying 
out  and  cancelling, 


and  C,  /  C  =  (p  +  r)  /  (p  +  q  +  r  +  5). 

It  is  evident,  from  an  inspection  of  the  Kirchhoff  equations 
belonging  to  the  three  cases,  that  if  the  resistances  of  the 
linear  conductors  which  go  to  make  up  a  given  network  are 
fixed,  and  if  Clt  C2,  C3)  •••  are  the  currents  in  the  different 
members  when  these  members  contain  the  electromotive  forces 
EU  E2,  Es,  -  •  •  and  C/,  C2f,  CB',  •  •  •,  the  corresponding  currents 
when  the  electromotive  forces  are  E±,  E2',  E^  •  •  -,  Cl  -\-  C/, 
C2  +  CJ,  C3  -\-  Cs',  •  •  -  would  be  the  currents  if  the  electromo 
tive  forces  were  El  -j-^',  E2  +  E2',  Ez  +  Ez',  .  .  .. 

Let  P  and  Q,  any  two  points  in  a  network  of  linear  con 
ductors  some  or  all  of  which  contain  electromotive  forces,  be 
at  potentials  VP,  VQ  respectively,  and  let  the  resistance  of 
the  whole  network  when  the  current  enters  at  one  of  these 
points  and  goes  out  at  the  other  be  r0,  then  if  P  and  Q  be 
connected  by  an  additional  wire  W  of  resistance  r,  the  cur 
rent  in  this  wire  will  be  (  VP  —  VQ)  /  (r0  +  r)  in  the  direction 
from  P  to  Q.  For  if  (1)  W  contained  an  electromotive  force 
(  VP  —  VQ)  directed  from  Q  to  P,  the  rest  of  the  network 


238  ELECTROKiNEMATICS. 

being  unchanged,  no  current  would  pass  through  W,  and  the 
other  currents  would  not  be  altered  by  the  introduction  of  W ; 
and  if  (2)  W  contained  the  electromotive  force  ( VP  —  VQ) 
directed  from  P  to  Q,  and  if  all  the  other  electromotive  forces 
in  the  original  network  were  annihilated,  leaving  the  resist 
ances  unchanged,  a  current  (  Vr  —  VQ}  /  (r0  +  ?')  would  flow 
through  W  from  P  to  Q :  the  given  arrangement  can  be 
regarded  as  formed  by  superposing  case  (1)  upon  case  (2). 

74.  The  Heat  developed  in  a  Circuit  which  carries  a 
Steady  Current.  Given,  in  a  region  not  exposed  to  magnetic 
changes,  a  chain  of  n  conductors,  each  in  itself  homogeneous, 
and  at  a  uniform  temperature  throughout ;  let  a  portion  A  of 
the  surface  of  the  first  be  kept,  by  means  of  some  external 
agency,  at  potential  VA,  and  a  portion  B  of  the  surface  of 
the  last  at  a  lower  potential  VR,  while  the  rest  of  the  outer 
surface  of  the  chain  abuts  upon  non-conducting  media.  Skt  k+l, 
the  surface  of  separation  between  the  &th  and  the  (k  +  l)th 
conductors,  may  or  may  not  be  equipotential,  but  if  these 
conductors  are  of  different  materials,  we  must  expect  to  find 
at  all  points  of  this  surface  a  uniform  discontinuity,  Ek >A  +  i> 
of  potential.  In  following  down  from  A  to  B  an  infinitesimal 
tube  of  flow  which  carries  the  steady  current  A  (7,  we  start  at 
potential  VA,  leave  the  first  conductor  at  potential  F/',  enter 
the  second  conductor  at  potential  F2',  leave  it  at  F"2",  enter  the 
third  conductor  at  F3',  and  so  on.  Every  second  in  the  kfh 
conductor,  A  C  absolute  units  of  electricity  are  lowered  from 
potential  Vk'  to  potential  FA;"  and  &C(Vk'  —  Vk")  units  of 
work  (representing  loss  of  electrostatic  energy)  are  done  by 
the  electrostatic  field  upon  the  electricity  which  moves  with 
the  current :  this  energy  appears  as  heat  in  this  conductor. 
The  work  thus  done  in  the  whole  chain  is 

A<7(F.t  -  F/'  +  F;  -  TV  +  JY  ~  V"  +  ' '  •  +  F.1  -  VB), 
or,  since  P*  +  i'  ~~  ^V  =  Ekt  *  +  i> 

\C(VA  -  rB-Mi,8  +  E^  +  •  •  •  +  ^H-i,w)  =  A6'(F4  -  VK  4-  E). 


ELECTROKINEMATICS.  239 

This  energy  all  appears  as  heat  in  the  conductors  which  form 
the  chain. 

At  the  surface  Sttt+l,  AC  units  of  electricity  are  raised 
every  second  from  potential  Vkn  to  potential  Vk+1'.  The 
work  thus  done  every  second  is  ^C-Ettk  +  lf  and,  by  virtue 
of  similar  processes  at  all  the  surfaces  of  discontinuity,  the 
electrostatic  energy  is  increased  in  this  way  every  second  by 
\C  •  E.  The  net  loss  in  electrostatic  energy  in  the  chain  per 
second  is,  therefore, 


which  is  otherwise  evident.  Taking  into  account  all  the  cur 
rent  filaments  which  go  to  form  the  steady  current  C,  we  see 
that  an  amount  of  energy  equivalent  to  C  (  VA  —  VB  -f-  E) 
appears  as  heat  in  the  conductors  which  form  the  chain,  and 
that  an  amount  of  electrostatic  energy  equal  to  EC  is  fur 
nished  to  the  chain.  If  the  chain  is  closed  and  if,  going 
around  it  in.  the  direction  of  the  steady  current  C,  we  denote 
by  E  the  algebraic  sum  of  the  discontinuities  of  potential, 
counting  a  step  up  as  positive,  we  shall  find  that  the  energy 
EC  appears  as  heat  in  the  conductors  and  that  since  the 
circuit  is  at  the  same  temperature  throughout,  this  is  fur 
nished  by  chemical  action  in  the  chain.  If  r  is  the  total 
resistance  of  the  chain,  C  =  E/  rand  EC  =  C2r.  This  result 
represents  ergs  or  joules,  according  as  E,  C,  and  r  are  meas 
ured  in  absolute  electrostatic  units  or  in  volts,  amperes,  and 
ohms  :  a  joule  is  equivalent  to  107  ergs. 

If  the  chain  contains  a  battery  of  electromotive  force  E0  in 
the  direction  of  the  steady  current  C,  and  if  there  are  in  the 
chain  outside  the  battery  discontinuities  of  potential  which. 
reckoned  against  the  current,  amount  algebraically  to  J5", 

E=E,-E',   C  =  (E,  -  E')  /  r, 

and  the  energy  used  in  heating  the  chain  is  (E0  —  E')  C  =  C*r  : 
when  we  wish  to  regard  the  battery  as  the  source  of  this 
energy,  it  is  convenient  to  write  the  last  equation  in  the 


240  ELECTROKINEMATICS. 

form  EfiC  =  C'2r  +  JK'C,  and  to  say  that  of  the  whole  energy, 
EQC,  furnished  by  the  battery,  (7V,  which  appears  as  heat  in 
the  conductors  which  form  the  circuit,  is  used  in  maintaining 
the  current,  and  IS'C,  in  overcoming  the  counter-electromotive 
force  E'.  If  a  cell  of  electromotive  force  EQ  be  joined  up  with 
a  number  of  metallic  conductors  all  at  the  same  temperature 
to  form  a  simple  circuit  of  total  resistance  r,  the  current  will 
be  CQ  =  EQ/r,  and  the  whole  energy,  EQCQ—C^rt  furnished 
each  second  by  the  battery,  will  appear  as  heat  in  the  circuit. 
If,  however,  while  the  total  resistance  of  the  circuit  remains 
unchanged,  the  battery  be  called  on  to  do  each  second  an 
amount  W  of  outside  work  of  any  kind  (such,  for  instance,  as 
that  involved  in  decomposing  an  electrolyte  in  the  external  cir 
cuit),  the  steady  current  will  have  a  value  C  smaller  than  COJ 
the  whole  energy  EQC  furnished  each  second  by  the  cell  will 
be  a  fraction  of  E0C0,  and  the  portion  of  it  (72r,  which  appears 
as  heat  in  the  circuit,  a  smaller  fraction  of  C(fr.  The  differ 
ence  between  EQC  and  C2r  will  be  equal  to  W,  and  this 
equation  determines  C. 

If  a  given  steady  current  C  is  to  be  conveyed  partly  by  a 
conductor  of  resistance  rl  and  partly  by  a  parallel  conductor 
of  resistance  r2,  and  if  the  portions  carried  by  these  conductors 
are  d  and  C2  respectively,  the  amount  of  heat  developed  per 
second  in  the  conductors  will  be  u  =  C^i\  +  C/r2.  If  Clt  and 
consequently  C2t  be  changed  so  as  to  keep  their  sum  equal  to 
the  constant  C,  u  will,  in  general,  change,  and  we  shall  have 
Dcu  =  2  Cj\  +  2  C2r,  •  DCC,  =  2  (  C.r,  -  <72ra) : 

u,  which  is  sometimes  called  the  dissipation  function,  will, 
therefore,  be  a  minimum  if  the  current  is  divided  between  rx 
and  r2  as  it  would  be  if  the  conductors  were  connected  at  the 
ends.  It  is  easy  to  prove  that  if  a  given  steady  current  be 
led  into  a  given  network  of  metallic  conductors,  at  a  uniform 
temperature,  from  without,  the  distribution  of  this  current  in 
the  network  will  be  such  as  to  make  the  dissipation  function 
as  small  as  possible.  If,  for  instance,  a  steady  current  C  be 


ELECTROKINEMATICS.  241 

led  into  the  network  represented  by  ABDF  in  Fig.  58  at  the 
point  A  and  out  again  at  B,  we  have 

cr  =  c-c,,  cp  =  c,  +  c-c,,  c,  =  c.-c, 

and  u  is  equal  to 

c-  cy  +  q(C9-  cgy  +  r(c-  cy  +  s-c*  +  ff.cg*. 


If  we  equate  to  zero  the  partial  derivatives  of  u  with  respect 
to  Cs  and  Cg,  we  shall  get  two  necessary  conditions  for  a 
minimum  :  the  equations  thus  obtained  are 

(P  +  9  +  ?)  Og  -  (p  +  q)  Cs  =  -pC, 
-(p  +  *)e,+(jP+t  +  r  +  *)C.*=(j»  +  r)  C, 

whence 

Cg/C  =  (qr  —  ps)  /  (gq  -f  sp  +  sg  +  sq  +  pg  +  rp  +  rg  -f  rq), 
C  /C=(gp  +  rp  +  rg  +  rq)  /  (gq  +  sp  +sg  +  sq  +pg  +  rp  +  rg  +  rq), 


etc.,  which  are  equivalent  to  equations  already  found. 

If  the  conductors  r^  r2,  ?*3,  •  •  •  rn  which  form  any  network, 
complete  or  not,  and  carry  currents  Cv  C2,  CSJ  —  •  Cn,  contain 
electromotive  forces  JEU  E2,  E&  -••  En  which  have  the  direc 
tions  assumed  for  the  currents,  the  currents  are  such  as  to 
make,  not  the  dissipation  function,  but 

W  =u-(  C,E,  -f  C2E,  +  C3E3  •  •  •  CHEn) 
a  minimum.     In  the  case  of  the  complete  Wheatstone's  Net, 


and  the  equations  formed  by  equating  to  zero  the  partial 
derivatives  of  W  with  respect  to  Cg,  Cs,  and  Cr  yield  the 
values  for  the  currents  given  in  the  last  section. 

75.  Properties  of  the  Potential  Function  inside  Conductors 
which  carry  Steady  Currents.  If  at  any  time  t,  positive  elec 
tricity  is  passing  through  a  linear  conductor  in  one  direction 
at  the  rate  P,  and  negative  electricity  in  the  other  direction 


242  ELECT  HOKINEMATICS. 

at  the  rate  N,  the  current  strength  is  P  -f-  N  in  the  first  direc 
tion.  Since  there  is  no  free  electricity  inside  a  homogeneous 
conductor  which  carries  what  we  have  called  a  steady  current, 
it  is  customary  to  assume,  when  one  uses  the  language  of  the 
"  Two  Fluid  Theory,"  that  such  a  current  consists  of  a  flow 
of  positive  electricity  in  one  direction  at  every  point,  and  an 
equal  flow  of  negative  electricity  in  the  opposite  direction. 
We  shall  avoid  much  circumlocution,  however,  and  we  shall 
introduce  no  error  into  our  numerical  computations  if  we 
speak  as  if  the  whole  current  were  due  to  the  motion  of  posi 
tive  electricity.  If  the  value  of  the  potential  function  within 
a  conductor  which  bears  a  steady  current  is  given,  all  the  cir 
cumstances  of  the  flow  in  the  conductor  are  fixed.  Positive 
electricity  flows  into  the  conductor  from  without  through  all 
parts  of  the  surface  where  the  derivative  of  the  potential 
function,  taken  in  the  direction  of  the  exterior  normal,  is  posi 
tive,  and  out  of  it  through  all  parts  of  the  surface  where  this 
derivative  is  negative.  At  all  points  where  the  conductor 
abuts  on  an  insulating  medium,  the  derivative  is  zero :  it  may 
be  zero  at  other  points  also.  There  can  be  no  closed  equi- 
potential  surface  lying  wholly  inside  a  conductor  which  carries 
a  steady  current,  unless  there  is  some  constant  source  of  posi 
tive  or  of  negative  electricity  within  this  surface,  for  the 
whole  flow  of  electricity  algebraically  considered,  per  unit  of 
time,  through  such  a  surface  from  within  outwards,  is  equal 
to  k  times  the  surface  integral  of  the  intensity  of  the  com 
ponent  of  force  in  the  direction  of  the  exterior  normal,  and 
this  is  not  zero.  There  must  then  be  such  a  constant  source 
of  free  electricity  within  the  surface  as  shall  furnish  just  as 
much  per  unit  of  time  as  the  current  carries  away. 

Although  it  is  not  very  easy  to  prove  analytically  that  — 
given  a  homogeneous  conductor  and  certain  portions  A,  B  of 
its  surface  which  are  to  be  kept  at  potentials  VA,  VB,  while  at 
all  other  portions  the  value  of  the  derivative  of  the  potential 
function  taken  in  the  direction  of  the  exterior  normal  is  to 
be  zero  —  there  exists  a  function  which  (1)  satisfies  these 


ELECTKOKINEMATICS.  243 

surface  conditions,  and  which  (2)  inside  the  conductor  satis 
fies  Laplace's  Equation,  and  with  its  first  space  derivatives  is 
continuous  and  single-valued,  it  is  nevertheless  clear  from 
physical  considerations  that  one  such  function  exists,  namely, 
the  potential  function  inside  the  conductor  when  A,  B  are  kept 
at  the  given  potentials  and  the  rest  of  the  surface  is  exposed 
to  an  insulating  medium.  For  practical  purposes  we  need  to 
prove  that  this  is  the  only  function  which  satisfies  the  given 
conditions.  Suppose  for  the  sake  of  argument  that  two  such 
functions,  V  and  W,  exist,  and  call  their  difference  u.  The 
function  u,  then,  satisfies  condition  (2)  and  is  itself  equal  to 
zero,  or  else  has  its  derivative  in  the  direction  of  the  exterior 
normal  equal  to  zero  at  every  point  of  the  surface.  Applying 
Green's  Theorem  in  the  form  of  Equation  151  to  u,  we  find 
that  the  quantity  (DxuY  +  (Dyu)*  -f  (Dsu)*,  which  can  never 
be  negative,  must  be  zero  at  every  point  within  the  conductor, 
so  that  Dxu,  Dyu,  and  Dzu  must  vanish  and  u  be  a  constant 
throughout  the  space  within  the  surface.  Xow  at  portions 
of  the  surface  itself,  u  is  zero,  hence  it  must  be  equal  to  zero 
everywhere  inside  the  conductor,  and  V—  W.  If  by  any 
means,  then,  we  find  a  function  which  satisfies  the  surface 
conditions  and  the  general  space  conditions  characteristic  of 
the  potential  function  inside  a  certain  conductor  carrying  a 
steady  current  under  given  surface  conditions,  this  function  is 
itself  the  potential  function. 

Any  surface  supposed  drawn  in  a  conductor  which  carries 
a  steady  current  in  such  a  way  that  the  derivative  of  the 
potential  function  taken  normal  to  this  surface  is  zero  shall 
be  called  a  surface  of  flow. 

If  a  conductor  which  under  given  surface  conditions  carries 
a  steady  current  be  cut  in  two  by  means  of  a  surface  of  flow, 
and  if  the  two  parts  be  separated  while  the  surface  conditions 
on  what  was  the  bounding  surface  of  the  old  conductor  remain 
the  same  as  before,  and  the  fresh  surfaces  now  abut  on  an 
insulating  medium,  the  state  of  flow  at  every  point  inside  each 
part  of  the  conductor  will  be  just  the  same  as  before,  for  the 


244  ELECTROKINEMATICS. 

values  of  V  and  Dn  V  on  the  surface  of  the  new  conductors  are 
what  they  were  before  separation,  and  V  must  have  its  old 
values  at  all  inside  points. 

When  a  conductor  is  cut  in  two  by  a  surface  of  flow  the 
fresh  surfaces  exposed  receive  a  statical  charge  of  free  elec 
tricity,  and  the  charges  on  what  was  the  bounding  surface 
of  the  original  conductor  are  in  part  changed  so  that  it  is 
only  within  the  parts  of  the  old  conductor  that  the  effect 
of  the  separation  is  nil  after  the  currents  have  become  again 
steady. 

If  two  mutually  exclusive  closed  surfaces  Sl  and  S2,  kept, 
respectively,  at  uniform  potentials  Vl  and  F"2,  are  the  elec 
trodes  of  an  infinite  homogeneous  conductor  7f,  of  specific 
conductivity  k,  which  fills  all  space  outside  these  surfaces 
and  is  at  potential  zero  at  infinity ;  if,  moreover,  the  steady 
flow  outward  through  ^  or  inward  through  S2  is  equal  to  C, 
the  current  vector  in  K  is  everywhere  equal  to  what  the  elec 
trostatic  force  would  be  if  K  were  air  and  if  Sl  and  S.2  had 
charges  C/kirk  and  —  C/ktrk  so  distributed  as  to  bring  them 
to  potentials  Vl  and  F2  respectively. 


In  most  of  the  preceding  discussion  we  have  tacitly  assumed 
the  separate  conductors  considered  to  be  homogeneous,  and  we 
shall  continue  to  do  so  in  the  following  sections  unless  the 
contrary  is  stated.  We  have  to  consider  briefly,  however,  in 
the  remainder  of  this  section  isotropic  conductors  which  have 
in  different  parts  different  specific  resistances. 

If  the  specific  conductivity  k  of  an  isotropic  conductor 
which  carries  a  steady  current  can  be  represented  by  a  posi 
tive  scalar  point  function,  and  if  the  components,  parallel  to 
the  coordinate  axes,  of  the  vector  q  which  represents  the 
current  strength,  are  u,  v,  and  w,  we  may  state  the  fact  that 
there  is  no  growing  accumulation  of  free  electricity  in  any 
portion  of  the  conductor  bounded  by  the  surface  S  by  the 
equation 


(   \ 


ELECTROK1NEMATICS.  245 

I    I  q  cos  (q,  n)  dS  =   I    (  q  [cos  (x,  n)  •  cos  (x,  q) 

+  cos  (y,  n)  -  cos  (y,  q)  4-  cos  (2,  w)  •  cos  (s,  q)~\dS 
[u  cos(#,  n)+  v  eos(y,  ?z)  +  w  cos  (2,  7i)]rf£ 

=   C  C  C[Dxu  +  ZV'  +  Dzw\dxdydz  =  0. 

Here  the  double  integrals  are  to  be  extended  over  the  whole 
of  S,  and  the  triple  integrals  over  all  the  space  included  by  S. 
Since  S  is  arbitrary,  the  integrand  of  the  triple  integrals  must 
be  equal  to  zero  at  every  point  within  the  conductor,  so  that 

Dtu  +  Dyv  +  Dzw  =  0  [198] 

and  q  is  a  solenoidal  vector. 

At  every  point  within  the  conductor, 


u  =  -        x,  v  =  -        y 
so  that 

2(k.DzV)=Q,    [199] 


or    k.\2V  +  (D3k.DxV  +  Dyk.DyV+Dzk-DzV)=Q.    [200] 

If  k  is  constant,  V  satisfies  Laplace's  Equation,  and  in  this  spe 
cial  case,  as  we  already  know,  none  of  the  free  electricity  which 
gives  rise  to  the  potential  function  V  is  within  the  conductor. 
Given  an  analytic,  scalar,  positive  point  function  k  and  a 
closed  analytic  surface  S,  it  is  easy  to  prove  by  the  help  of 
[149]  that  there  cannot  be  two  different  functions,  Fi  and  F"2, 
which  (1)  with  their  first  derivatives  are  continuous  within  S 
and  at  every  point  in  this  region  satisfy  the  equation 


(2)  on  the  given  portions  Si  and  S2  of  S  have  at  each  point 
equal  values,  and  (3)  on  the  rest  of  S  have  at  every  point 
equal  normal  derivatives. 

The  differential  equations  of  the  current  lines  are 

dx  _  dy  _  dz 
u         v         w 


246  ELECTKOKINEMATICS. 

At  a  surface  of  separation  between  two  conductors  which 
carry  a  steady  current  the  normal  components  of  the  current 
and  the  tangential  components  of  the  electrostatic  force  are 
continuous.  If  0X  and  02  are  the  angles  which  the  resultant 
electrostatic  forces  f\  and  F2  make  with  the  normal  on  the 
two  sides  of  such  a  surface  at  any  point, 

ktFi  cos  Oi  =  k2F2  cos  02  and  F:  sin  6l  =  F2  sin  02, 
whence,  by  dividing  the  members  of  the  first  of  these  equations 

by  the  corresponding  members  of  the  second,  — - — -  =  — - — -, 

K\  K2 

an  equation  which  shows  how  the  current  lines  are  refracted 
at  the  surface.  At  a  surface  of  separation  between  copper 
and  manganin  where  the  ratio  of  the  conductivities  is  about  30, 

0!  =  27°  42'  when  02  =  1°,  and  0L  =  69°  09'  when  62  =  5°. 

If  %  and  n2  represent  normals  drawn  from  any  point  of  the 
surface  of  separation  between  two  conductors  which  are  carry 
ing  a  steady  current  into  the  first  and  the  second  conductor 
respectively, 

klDniYl  +  k2Dn2V=0.  [201] 

76.  Method  of  finding  Cases  of  Electrokinematic  Equilib 
rium,  If  w  is  a  single-valued,  generally  continuous  solution 
of  Laplace's  Equation,  Aw  -\-  B,  where  A  and  B  are  constants, 
is  another  such  function  which  has  the  same  level  surfaces 
as  w.  If  an  area  be  chosen  on  one  of  these  surfaces,  it  is  pos 
sible  to  draw  through  every  point  of  its  perimeter  a  line, 
defined  by  the  equations  dx  /  Dxw  =  dy  /  Dyw  =  dz/Dzw,  which 
shall  cut  orthogonally  all  the  level  surfaces  of  w  which  it 
meets.  All  these  lines  form  a  tubular  surface  such  that  the 
normal  derivative  of  w  at  every  point  of  it  is  zero.  If  I7  is  a 
portion  of  space  bounded  by  such  a  tube  and  by  portions,  Sf,  S", 
of  two  of  w's  level  surfaces  on  which  it  has  the  values  w1  and 
w"  respectively,  w  is  identical  with  the  potential  function  that 
would  govern  the  flow  within  any  homogeneous  conductor  of 
the  form  T  if  the  surface  S'  and  S"  were  kept  at  potential  w' 


ELECTROKINEMATICS.  247 

and  w",  while  the  rest  of  the  boundary  was  a  surface  of  flow. 
Moreover,  Aw  •+•  B,  where  A  and  B  can  be  chosen  at  pleasure, 
must  be  the  potential  function  within  a  homogeneous  conduc 
tor  of  the  form  T,  if  the  surface  S'  and  S"  were  kept  at  poten 
tials  Aiv'  -f  B,  Aw"  +  B  respectively,  the  rest  of  the  boundary 
being  a  surface  of  flow.  By  using  different  pairs  of  level  sur 
faces  of  w  and  tubes  of  different  forms,  it  is  possible  with 
the  help  of  this  one  function  to  study  the  laws  of  steady  flow 
inside  conductors  of  many  different  shapes  and  to  obtain 
results  some  of  which  may  happen  to  be  practically  inter- 

£ 

esting.      For  instance,  w  =  -  -f-  d,  where  c  and  cl  are  constants 

and  r  the  distance  from  a  fixed  origin  0  to  the  point  (a*,  y,  «), 
gives  the  value  of  the  potential  function  inside  a  conductor 
bounded  by  two  spherical  surfaces  of  radii  a  and  b  having  0 
as  their  common  centre  when  these  surfaces  are  kept  respec- 

C-  G 

tively  at  potentials  -  +  d  and  -  +  d.     In  this  case  the  whole 

amount,  per  unit  of  time,  of  positive  electricity  which  enters 
the  conductor  through  the  surface  r  —  a  crosses  every  equi- 
potential  spherical  surface  within  the  conductor  and  leaves  it 
by  the  surface  r  =  b  is  4  trek,  where  k  is  the  specific  conduc 
tivity  of  the  material  out  of  which  the  conductor  is  made. 
The  resistance  of  the  conductor  is,  by  definition, 

c  _  c 

a      b       b  —  a 


4  -n-ck      4  -n-kab 

a  quantity  independent  of  c  and  d. 

It  is  evident  that  any  conical  surface  the  vertex  of  which  is 
0  will  be  in  this  case  a  surface  of  flow,  and  that  the  function 

w  =  -  +  d  governs  the  flow  in  any  piece  cut  out  of  the  spheri 
cal  shell  just  considered  by  such  a  surface.  It  is  easy  to  see 
that  if  w  is  the  solid  angle  of  the  cone,  the  resistance  of  the 

portion  of  the  conductor  cut  out  will  be  -    —  • 

kuab 


248  ELECTKOKINEMATICS. 


(1         1 
=  c[ 

Vl          ?*2 


Again,  the  equation  V=  c( )  +  d,  where  r±  and  r2  are 

VI  r2/ 

the  distances  of  the  point  (x,  y,  z)  from  the  fixed  points  Ox 
and  02,  gives  us  the  potential  function  inside  an  infinite 

conductor  bounded  in  part  by  the  surfaces =  a  and 

T\          r2 

=  b,  when  the  first  is  kept  at  potential  ac  +  d,  the 

r\      r2 

second  at  potential  be  +  d.  In  this  case  the  surface  V  =  d 
is  a  plane  bisecting  at  right  angles  the  straight  line  0^0^ 
Larger  and  smaller  values  of  V  than  this  give  closed  surfaces, 
each  of  which  surrounds  one  of  the  points  and  leaves  the 
other  outside.  For  very  large  values  of  V,  if  c  is  positive, 
the  equipotential  surfaces  are  very  small,  nearly  spherical 
surfaces  surrounding  0^ 

To  find  the  amount  of  positive  electricity  which  enters 
the  conductor  under  consideration,  per  unit  of  time,  through 
the  surface  V  =  ac  +  d,  where  ac  shall  be  positive,  we  must 

integrate  over  this  surface  —  kDnV  or  —  kc\  Dn(  —  \  —  Dn—    . 

L       \ri/  r2j 

According  to  Green's  Theorem,  the  resulting  integral  is  exactly 
the  same  as  that  taken  over  any  other  closed  surface,  large  or 
small,  which  surrounds  0:  and  leaves  02  outside.  Let  us 
consider,  then,  a  spherical  surface  of  radius  e  <  0L0.,  whose 
centre  is  at  Oj.  The  required  integral  in  this  case  is  —  4  -rr^k 
times  the  average  value  of  Dr  V  taken  over  the  spherical 
surface  ;  or,  since  ^  for  all  points  on  this  surface  is  equal  to  e, 

4  7T€2kc    —  —  average  value  of  Dr  (  —  )     • 


•.a) 


If,  now,  e  be  made  smaller  and  smaller,  Dr  (  —  ]  always  has 

1  V2/ 

some  finite  value  for  every  point  on  the  surface  of  the  sphere 
surrounding  0,,  and  the  expression  just  given  approaches  the 
limiting  form  4  -n-kc.  Hence,  4  -n-kc  units  of  positive  electricity 


ELECTROKINEMATICS.  249 

enter  the  given  conductor  through  the  surface  V  =  ac  +  d  in 
every  second,  whether  this  surface  is  large  or  small.  The 
resistance  of  the  conductor  between  the  surfaces  V  =  ac  -f  d 

and  V  =  be  +  d  is,  by  definition  of  the  term,  — — —  • 

4  7T/C 

If  a  and  b  are  made  very  large  and  equal,  with  opposite 
signs,  the  two  surfaces  through  which  electricity  enters  and 
leaves  the  conductor  become  very  nearly  coincident  with 

spherical  surfaces  of  radius  e  =  —  drawn  about  0±  and  02 
respectively.  The  resistance  of  the  conductor  in  this  case  is 
- — —  •  Considerations  of  symmetry  show  that  any  plane  which 

^  TTK€ 

contains  the  line  0L02  is  a  surface  of  flow.  If  we  cut  the 
conductor  in  two  by  such  a  plane,  we  shall  have  an  infinite 
conductor  with  two  nearly  hemispherical  electrodes  sunk  in 
its  plane  surface.  The  resistance  of  this  part  of  the  whole 

conductor  is  — — ,  a  quantity  independent  of  the  distance  apart 

7TrC€ 

of  the  electrodes.     This  is  nearly  the  case  of  two  poles  of  a 
battery  sunk  in  the  earth. 
Again,  the  expression 


where  rl  and  r2  are  the  distances  of  a  point  P  in  space  from 
any  two  parallel  straight  lines,  A  and  B,  is  a  solution  of 
Laplace's  Equation  which,  with  its  derivatives,  vanishes  at 
an  infinite  distance  from  these  lines  and  which  is  constant 
all  over  any  one  of  a  double  system  of  circular  cylindrical 
surfaces  (Fig.  59),  some  of  which  surround  one  of  the  given 
lines  and  some  the  other.  This  function,  then,  when  c  and  d 
are  properly  determined,  is  the  potential  function  within  an 
infinite  lamina,  either  thick  or  thin,  when  that  lamina  is  per 
forated  perpendicularly  to  its  plane  by  two  circular  cylindri 
cal  holes,  the  curved  surfaces  of  which  are  kept  at  given 


250 


ELECTROKINEMATICS. 


constant  potentials.  2  irkc  units  of  positive  electricity  per 
unit  of  time  per  unit  of  thickness  of  the  lamina  enter  the  con 
ductor  through  one  of  the  cylindrical  surfaces,  and  the  same 
amount  leaves  it  by  the  other  surface.  The  resistance  of  the 
lamina  is  then  the  difference  between  the  values  of  the  poten 
tial  function  at  the  electrodes  divided  by  2  irkc  times  the 
thickness  of  the  lamina. 

These  examples  will  serve  to  show  how  we  may  discover  an 
indefinite  number  of  cases  of  kinematic  equilibrium  by  assum 
ing  some  function,  in  general  finite  and  continuous,  which 


FIG.  59. 

satisfies  Laplace's  Equation,  and  then  taking  as  a  conductor 
one  inside  which  the  given  function  is  everywhere  finite,  and 
which  is  bounded  by  surfaces  over  each  of  which  either  the 
function  is  constant  or  its  normal  derivative  zero. 

If  we  transform  Equation  199  to  orthogonal  curvilinear 
coordinates  defined  by  the  scalar  point  functions  u,  v,  w, 
where  w  satisfies  Laplace's  Equation,  and  assume  V  to  be 
expressible  as  a  function  of  w  only,  we  shall  obtain  (see 
page  182)  the  equation  DJV+  DWV  •  Dwk  /  k  =  0.  If  the 
specific  conductivity  of  a  body  occupying  the  space  T  men 
tioned  at  the  beginning  of  the  section  were  not  constant  but 
a  given  function  of  w,  this  equation  would  determine  V. 


ELECTROMAGNETISM.  251 


HI.     ELECTROMAGXETISM. 

77.  Electromagnetism.  Straight  Currents.  If  a  steady 
electric  current  be  sent  through  a  long  straight  wire,  the  space 
in  the  neighborhood  of  the  current  becomes  a  field  of  magnetic 
force.  If  the  medium  about  the  conductor  is  homogeneous, 
the  direction  of  the  field  is  such  that  a  small  magnetic  needle 
freely  suspended  by  its  centre  tends  to  set  itself  perpendicular 
to  the  wire  and  to  the  perpendicular  dropped  from  the  point 
of  suspension  upon  the  wire,  so  that  "  if  a  person  be  imagined 
as  swimming  in  the  current  which  flows  from  his  feet  to  his 
head,  and  if  he  face  the  needle,  the  north  pole  will  be  turned 
towards  his  left  hand."  The  field  is  symmetrical  about  the 
wire  and,  according  to  the  rule  just  given,  its  direction  at  any 
point  is  normal  to  the  plane  drawn  through  the  point  and  the 
wire,  so  that  the  lines  of  force  are  circumferences  forming 
right-handed  whirls  about  the  current.  To  investigate  the 
law  of  the  change  of  the  intensity  of  the  force  with  the  dis 
tance  from  the  wire,  we  may  imagine  a  rigid  frame  free  to 
turn  about  the  vertical  wire  as  a  hinge,  and  suppose  a  magnet 
to  be  rigidly  attached  to  this  frame.  It  will  be  found  that 
in  this  case  the  frame  will  have  no  tendency  to  rotate  under 
the  action  of  the  electromagnetic  forces,  so  that  the  sum  of 
the  moments  about  the  wire,  of  the  forces  which  the  field 
exerts  upon  the  magnet,  must  be  zero.  If  i\  and  rz  are  the 
distances  of  the  poles  from  the  wire,  and  if  F(r)  is  the  inten 
sity  of  the  field  at  a  distance  r  from  the  wire,  the  equality  of 
moments  shows  that,  however  the  magnet  be  placed  on  the 
frame, 


or,  in  general,  r  •  F(r)  =  a  constant,  k.  The  value  of  k  is 
found  to  be  dependent  upon  the  strength,  C,  of  the  current  in 
the  wire,  and  can  be  used  to  define  this  strength.  We  may 
write,  therefore,  F(r)  =  A-  C/r,  where  A  is  a  constant  depend 
ing  upon  the  units  in  which  C  is  measured.  If  we  use  the 


252  ELECTROMAGNETISM. 

absolute  electromagnetic  c.g.s.  units,  defined  below,  in  deter 
mining  C,  it  will  presently  appear  that  A  is  2. 

If  we  take  the  plane  of  the  paper  for  the  xy  plane,  and 
imagine  the  wire  which  carries  the  current  to  cut  the  paper 
normally  at  the  origin,  then,  if  the  current  comes  from  below, 
the  components  of  the  field  at  the  point  (x,  y)  are 

X  =  -  2  C  sin  (x,  r)  /r  and    Y  =  2  C  cos  (x,  r)  /r, 
or         X  =  -  2(77// 02-fy2)   and   Y  =  2Cx/(x2  +  y2). 

Here  DVX  =  DXY  and  the  magnetic  force  is,  in  general,  a 
lamellar  vector,  so  that  it  has  a  potential  function  which,  since 
the  lines  of  force  are  closed,  must  be  multiple-valued.  This 
potential  function  is  evidently 

±26'  tan-1  (y/x)  +  constant,  or  ±  2  CO  +  constant, 

and  it  satisfies  Laplace's  Equation.  The  plus  or  the  minus 
sign  is  to  be  chosen  according  as  we  wish  to  use  the  derivative 
of  the  potential  function  taken  in  any  direction,  or  its  nega 
tive,  as  a  measure  of  the  component  of  the  field  in  that  direc 
tion.  The  line  integral  of  the  tangential  component  of  the 
force  taken  around  any  curve  in  the  xy  plane  which  sur 
rounds  the  origin  is  4?r(7,  so  that  we  infer  from  Stokes' s  The 
orem  that  at  the  origin  the  magnetic  force  is  not  lamellar.  If 
a  magnetic  pole  of  strength  m  be  moved 
around  any  closed  path,  the  work  done  on 
it  by  the  magnetic  field  will  be  4  irm  C  if 
the  path  link  right-handedly  once  with 
the  wire,  or  zero  if  the  path  do  not  link 
with  the  circuit.  These  results  are  found 
to  be  independent  of  the  inductivity  of  the 
homogeneous  medium  about  the  wire. 

Since   DXX  +  Dy  Y  =  0,    the   force   in 
the  medium  about  the  wire  is  solenoidal, 

and  the  whole  flux  of  force  from  within  outward  through  any 
closed  surface  is  zero.  If  two  straight  lines  parallel  to  the 
wire  are  distant  a  and  b  centimetres  from  it  respectively, 


ELECTROMAGNETISM.  253 

the  flux  of  force  (Fig.  60)  through  the  unit  length  of  any  cylin 
drical  surface  bounded  by  the  lines  is  2  C -log (b/a).  Since 
we  have  assumed  that  a  finite  quantity  of  electricity  is  carried 
by  a  conductor  of  zero  cross-section,  it  is  not  surprising  that  this 
useful  analytic  result  becomes  infinite  if  either  a  or  b  is  zero. 

If  two  infinitely  long  straight  wires  parallel  to  the  z  axis 
carry  equal  steady  currents  of  strength  C  in  opposite  directions, 


FIG.  61. 

and  if  they  cut  the  xy  plane  at  the  points  Ai}  A2,  which  have 
the  coordinates  (a,  0),  (—  a,  0)  respectively,  the  scalar  potential 
function,  O,  of  the  field  has  at  the  point  (x,  y,  z)  the  value 

2  C.tan-iO/t*  -  a)]  -  2  C •  tan- 1  [y / (x  +  «)], 
or  2  C  •  tan- l  [2  ay /  (x*  -f  if  -  a2)]. 

The  conjugate  function,  <l>,  is  +  2  C-logfa/rs),  where  i\  and 
>-2  are  the  distances  of  the  point  (a-,  y,  z)  from  A^  and  Az 


254  ELECTROMAGNET1SM. 

respectively.  The  lines  of  force  and  the  traces  in  the  xy 
plane  of  the  equipotential  surfaces  are  shown  in  Fig.  61. 

DXO  =  —  Dy$,  Dy£l  =  Dx$,  and  the  derivative  of  O  at  any 
point  in  the  xy  plane  taken  in  any  direction  in  the  plane  is 
equal  to  the  derivative  of  3>  at  the  same  point  taken  in  a  direc 
tion  in  the  plane  at  right  angles  to  the  first.  If,  then,  a 
curve  c  is  the  trace  in  the  xy  plane  of  a  cylindrical  surface  S, 
the  generating  lines  of  which  are  parallel  to  the  z  axis,  and  if 
n  represents  a  direction  in  the  plane  perpendicular  to  c,  the 
line  integral  of  DnQ,  taken  along  c  represents  the  flux  of  mag 
netic  force  across  S  per  unit  of  its  height,  perpendicular  to  the 
xy  plane.  This  integral  is  equal  to  the  line  integral  of  the 
tangential  derivative  of  <£  along  c  or  to  the  difference  between 
the  values  of  <I>  at  the  ends  of  the  curve.  If  this  difference 
is  nothing,  the  corresponding  flux  is  nothing ;  if  $  is  constant 
all  along  c,  this  curve  is  a  line  of  force. 

From  the  results  just  obtained,  it  is  evident  that  if  two 
straight  lines  parallel  to  the  z  axis  cut  the  xy  plane  in  the 
points  J51?  Bz  respectively,  the  flux  of  magnetic  force  through 
a  cylindrical  surface  bounded  by  these  lines,  per  unit  of  its 
length,  parallel  to  the  z  axis,  is 

2  C  •  log  [(^A  -  AZBZ)  I (A,B,  -  AiBj)-]. 

This  represents  the  flux  of  force  per  unit  of  its  height,  through 
a  circuit  sz,  consisting  essentially  of  two  infinitely  long  straight 
wires,  parallel  to  the  -z  axis,  cutting  the  xy  plane  at  B1}  Bz 
when  the  steady  current  C  traverses  the  circuit  s1?  consisting 
essentially  of  the  two  wires  already  mentioned,  which  cut  the 
xy  plane  at  AI  and  A2.  Symmetry  shows  that  this  expression 
would  also  give  the  flux  through  s1;  due  to  a  steady  current 
C  in  sz. 

If  an  infinitely  long  cylindrical  conductor,  the  generating 
lines  of  which  are  parallel  to  the  z  axis,  and  which  is  sur 
rounded  by  a  homogeneous  medium,  carry  a  steady  current  in 
the  direction  of  its  length,  and  if  the  current  density  at  the 


ELECTROMAGNETISM.  255 

point  (x',  y',  z')  be  </',  a  function  of  .*•'  and  y'  but  not  of  z',  the 
intensity  of  the  magnetic  field  //  within  or  without  the  con 
ductor  can  be  obtained  by  imagining  the  conductor  made  up 
of  separate  current  filaments,  each  of  which  has  a  field  like 
that  about  a  fine  straight  wire,  unaltered  by  the  presence  of 
the  others.  If  L,  M,  N  are  the  intensities  of  the  components 
of  If  parallel  to  the  coordinate  axes,  L  and  M  are  functions 
of  x  and  y  while  N  is  zero. 

_       r  rZg'ds-^dx'dy'  r  r  2  g'  (x  -  x')  dx'  dy' 

JJ  (x-x'T+(y-!/y2'  JJ   (x-x<y+(y-y<y' 

where  the  double  integrals  extend  over  the  section  of  the  con 
ductor  made  by  the  xy  plane.  If  the  whole  amount  of  cur 
rent  in  the  conductor  is  C,  and  if  u  represents  the  distance 
of  the  point  (x,  y,  z)  from  the  axis  of  «,  and  <f>  the  angle 
tan"1  (*//#),  uL  and  i(M  approach  the  limits  —  2  C  •  sin  <f>  and 
2  C  •  cos  <£  when  u  increases  without  limit.  The  line  integral 
of  the  tangential  component  of  the  field,  taken  around  any 
curve,  which  surrounds  the  conductor,  is  equal  to  the  corre 
sponding  integral  taken  around  a  circle  in  the  xy  plane  of 
infinite  radius,  with  centre  at  the  origin.  The  value  of  this 
last  integral  is  obviously  kirC.  Except  for  points  in  the 
mass  of  the  conductor,  the  integrands  of  the  expressions  for 
L  and  J/  are  continuous  functions  of  x  and  y  for  all  values  of 
x'  and  y'  within  the  limits  of  integration,  and  DyL  =  DXM  and 
D*L  +  DyM  =  0. 

At  all  points  in  empty  space  near  the  conductor,  therefore,  the 
field  is  solenoidal  and  lamellar  and  there  is  a  potential  function 


which  satisfies  Laplace's  Equation. 

In  the  special  case  where  the  conductor  is  in  the  form  of  a 
right  circular  cylinder  (or  of  concentric  shells  bounded  by 
cylindrical  surfaces  of  revolution),  and  where  the  current  den 
sity  is  a  function  only  of  the  distance  from  the  z  axis,  which 


256  ELECTKOMAGNETISM. 

coincides  with  the  axis  of  the  conductor,  the  field  is  evidently 
symmetrical,  and  the  direction  of  the  force  at  any  point  is 
perpendicular  to  the  perpendicular  to  the  axis  drawn  through 
the  point.  Everywhere  in  empty  space  in  the  vicinity  of  the 
conductor  a  potential  function,  O,  exists,  and,  since  Dr£l  =  0, 
Laplace's  Equation  degenerates  into  Z>/O  =  0,  or  O  =  aO  +  I. 
The  work  done  by  the  field  when  a  magnetic  pole  of  strength 
m  moves  around  a  circumference,  the  axis  of  which  is  the  z 
axis,  is  evidently  equal  to  ±.kirCm,  where  C  is  the  sum 
of  the  currents  in  all  the  current  filaments  which  the  path 
encloses.  Since  the  line  integral  of  Z>S,O  taken  around  any 
such  path  in  empty  space  in  right-handed  direction  around 
the  current  is  2  TT&,  a  is  equal  in  absolute  value  to  twice  the 
whole  current  carried  by  so  much  of  the  conductor  as  lies 
within  the  path.  If  the  direction  of  the  z  axis  is  such  that, 
if  the  eye  is  in  the  positive  x  axis  looking  at  the  origin,  a 
counter-clockwise  rotation  of  the  positive  axis  of  y  through 
90°  would  make  it  coincide  with  the  positive  z  axis,  and  if 
O  =  —  2  CO  -h  b,  the  force  at  any  point  not  in  the  mass  of  the 
conductor,  in  any  direction,  is  the  derivative  of  O  at  that  point 
taken  in  the  direction  in  question,  and  the  resultant  force  is 
—  Dfl  I  r  or  2  C  /  r.  This  is  the  same  as  if  all  the  current 
nearer  the  z  axis  than  the  point  in  question  were  flowing 
through  a  fine  wire  coincident  with  the  axis  of  z.  If  the 
infinitely  long  cylindrical  conductor  is  a  uniform  tube,  the 
axis  of  which  is  the  z  axis,  O  =  aO  -f-  b  in  the  empty  space 
within  the  tube,  and,  since  (on  account  of  symmetry)  the 
resultant  force  a/r  must  vanish  on  the  z  axis,  a  is  zero  and 
the  intensity  of  the  field  within  the  tube  is  everywhere  zero. 

We  may  easily  find  the  intensity  of  the  electromagnetic 
force  at  any  point  P  within  an  infinitely  long,  round  con 
ductor  carrying,  in  the  direction  of  its  length,  a  steady  cur 
rent  with  intensity  the  same  at  all  points  equally  distant 
from  the  axis  of  the  conductor,  if  we  imagine  a  cylindrical 
surface,  S,  of  revolution  coaxial  with  the  conductor  drawn 
through  P.  The  magnetic  force  at  P,  due  to  so  much  of  the 


ELECTROMAGNETISM.  257 

current  as  lies  outside  *S',  is  nothing;  the  force  due  to  so  much 
of  the  current  as  lies  within  S  is  evidently  the  same  as  if  this 
portion  of  the  current  were  concentrated  in  the  axis.  Tf,  there 
fore,  a  straight  conductor  in  the  form  of  an  infinitely  long 
cylinder  of  revolution  of  radius  a  carries  a  steady  current  C  in 
the  direction  of  its  length,  and  if  the  intensity  (q)  of  the  cur 
rent  is  a  function  only  of  the  distance  (r)  from  the  axis  of  the 
conductor,  the  intensity  of  the  magnetic  force  (Jf)  is  2C/r 

without  the  cylinder  and  -  -  I    xqdx  within.     The   flux  of 

V    */0 

induction  per  unit  length  of  the  cylinder  across  so  much  of 
any  plane  through  the  axis  as  lies  within  the  conductor  is 


Xa     cjr     f*r 
p—  I    xq  dx. 
)'  »/o 


If  q  does  not  involve  r,  the  current  is  uniformly  distributed 
through  the  conductor,  the  strength  of  the  field  within  the 
cylinder  is  2  CV/a2,  and  Q  is  equal  to  pC.  If  the  axis  of  the 
cylinder  is  the  z  axis,  the  force  components  at  any  inside  point 
distant  r  from  the  axis  are  L  =  —  2  Cy  /  a?,  M  =  2  Cx/a2,  so 
that  H  is  solenoidal,  as  it  would  be  if  q  were  any  analytic 
function  of  r.  Since  H  is  not  lamellar  within  the  conductor, 
it  is  at  the  outset  clear  that  there  can  be  no  scalar  poten 
tial  function  O  there ;  it  is  well  to  notice,  however,  that,  if 
the  derivative  of  a  scalar  function,  Q,  at  any  point  in  any 
direction  were  required  to  show  the  force  at  that  point  in  the 
given  direction,  it  would  need  to  satisfy,  within  the  conductor, 
the  two  incompatible  conditions, 

D,ti  =  0,     (D9Q)  /r=-2  Cr/a*. 

Since  H  is  solenoidal  even  at  inside  points,  we  may  ask 
whether  its  components  are  not  the  components  of  some  vector, 
Q,  which  may  be  regarded  as  a  vector  potential  function  of 
//,  and  it  is  clear  that  a  vector  of  intensity  —  u-gr2,  directed  at 
every  point  parallel  to  the  z  axis,  satisfies  all  the  conditions, 
as  do  many  other  vectors.  The  component,  at  any  point 
within  the  conductor,  in  any  direction,  of  the  curl  of  the 


258 


ELECTROMAGNETISM. 


vector  (0,  0,  —  Tr^r2)  shows  the  component  of  the  magnetic 
force  //  at  the  point  in  the  given  direction.  The  abscissas 
of  Fig.  62  represent  distances  from  the  axis  of  the  conductor, 


FIG.  62. 

and  the  ordinates  the  corresponding  values  of  the  resultant 
magnetic  force  in  the  case  just  considered. 

If  a  uniformly  distributed  current  C  be  brought  up  normally 
through  the  plane  of  the  paper  by  an  infinitely  long  cylinder 
of  revolution  and  down  through  a  similar  cylinder  parallel  to 
the  first,  the  lines  of  force  without  the  cylinders  are  of  the 
same  shape  as  those  shown  in  Fig.  61.  The  curve  in  Fig.  63 
shows  the  intensity  of  the  field  at  points  in  a  straight  line 
which  cuts  the  axes  of  the  cylinders  perpendicularly. 

If  two  infinitely  long,  coaxial,  cylindrical  surfaces  of  revo 
lution  carry  symmetrically  equal  and  opposite  currents,  each 


FIG.  63. 


of  strength  C,  parallel  to  their  common  axis,  the  space  between 
the  surfaces  is  a  field  of  electromagnetic  force  of  strength 
2  C /r,  where  r  is  the  distance  from  the  axis.  There  is  no 
force  within  the  inner  surface  or  without  the  outer  one. 


ELECTROMAGNETISM.  259 

In  the  case  of  a  long,  straight  wire  of  radius  a  surrounded 
by  a  coaxial  tube  of  radii  b  and  c,  and  carrying  uniformly 
distributed  a  steady  current  C  which  returns  through  the 
tube,  the  electromagnetic  force  is  evidently  zero  on  the  axis 
of  the  wire  and  continuous  at  every  distance  r  from  the  axis. 
If  Wi  and  w2  are  the  intensities  of  the  current  in  the  wire 
and  in  the  tube  respectively,  C  =  w^a2  =  u\ir  (c2  —  #2),  and 
if  we  apply  the  formulas  just  proved,  we  shall  learn  that 
the  strengths  of  the  fields  writhin  the  wire,  between  the  wire 
and  the  tube,  in  the  body  of  the  tube  and  without  the  tube, 
are  given  by  the  expressions  2  Trw-p,  2  Tra?u\/r,  2  TTWZ  (c2  —  r2)/r, 
and  0. 

It  is  to  be  noted  that  the  strength  of  the  magnetic  field  due 
to  a  given  electric  current  is,  in  the  homogeneous  medium 
which  surrounds  the  current,  wholly  independent  of  the  per 
meability  of  this  medium,  whereas  the  field  due  to  a  given 
magnet  would  be  inversely  proportional  to'  the  inductivity. 
If  the  fields  of  a  given  circuit  and  a  given  magnet  were  the 
same  in  one  homogeneous  medium,  they  would  not  be  the 
same  in  another  homogeneous  medium  of  different  magnetic 
inductivity.  The  induction  due  to  a  current  circuit  in  a  homo 
geneous  medium  filling  all  space  is  proportional  to  the  induc 
tivity,  as  is  the  energy  in  the  medium.  The  induction  due  to 
magnetic  matter  surrounded  by  a  homogeneous  medium  is 
independent  of  the  inductivity  of  the  medium.  The  action 
of  a  distribution  of  magnetic  matter  in  an  infinite  homogene 
ous  medium  on  a  circuit  carrying  a  steady  current  is  not 
altered  by  changing  the  inductivity  of  the  medium. 

78.  Closed  Circuits.  Experiment  shows  that  if  a  steady 
current  of  strength  C  runs  in  a  simple  linear  circuit  of  any 
form,  there  is  a  magnetic  field  in  the  neighborhood  of  the 
conductor  and  the  lines  of  the  field  are  all  linked  right- 
handedly  with  the  circuit.  If  a  unit  magnetic  pole  be  carried 
round  any  closed  path  which  does  not  link  with  the  circuit, 


260 


ELECTROMAGNETISM. 


FIG.  64. 


the  work  done  by  the  field  on  the  pole  is  zero,  whatever  the 
character  of  the  medium  near  the  circuit,  so  that  a  potential 
function  exists  in  the  so-called  empty  space  about  the  wire. 
This  potential  must  be  multiple-valued,  since  the  lines  of  force 
are  closed.  If  the  pole  be  carried  round  a  closed  path  which 
links  once  with  the  circuit,  the  work  done  on  the  pole  by  the 
field  is  ±  4  TT  (7,  whether  the  medium 
intersected  by  the  path  is  homogeneous 
or  not.  We  infer,  therefore,  that  no 
scalar  potential  function  exists  in  the 
wire  which  carries  the  current. 

It  follows  from  the  experiments  of 
Ampere  that  the  field  of  magnetic  force, 

due  to  a  steady  current  of  C  electromagnetic  units  flowing 
in  a  closed  linear  circuit  in  a  homogeneous  medium,  is  iden 
tical  with  the  field  of  magnetic  Induction  due  to  a  simple 
magnetic  shell  (Fig.  64)  of  strength  C  bounded  by  the  circuit. 
This  statement  defines  the  electromagnetic  unit  of  current. 
The  magnetic  force,  due  to  a  current  of  C  electromagnetic 
units  flowing  in  a  closed  linear  circuit  in  a  homogeneous 
medium  of  inductivity  /x,  is  the  same  in  magnitude  and  direc 
tion  at  any  point  P  as  the  force  due 
to  a  simple  magnetic  shell  of  strength 
C/A  bounded  by  the  circuit..  The  shell 
may  be  of  any  form,  provided  that  it 
does  not  pass  through  P  and  that  its 
positive  side  is  such  that  the  current 
surrounds  right-handedly  the  direc 
tion  of  polarization.  To  make  the 
potential  function  single-valued,  we 
may  cover  the  circuit  by  a  cap  or  dia 
phragm,  fix  at  pleasure  the  value  O0 

of  the  potential  function  at  some  one  point  0  in  the  field,  and 
define  the  value  at  any  other  point  Q  to  be  the  line  integral  of 
the  magnetic  force  taken  from  0  to  Q  along  any  path  which 
does  not  cut  the  diaphragm. 


FIG.  65. 


ELECTROMAGNETISM . 


261 


At  any  point  P  on  the  axis  of  a  circular  current  of  radius 
a,  at  a  distance  x  from  the  plane  of  the  circuit,  the  circuit 
subtends  the  solid  angle 

o>  =  27r(l  -  cos  ff)  =  2  TT  (1  -  x  /  Va2  +  x2). 

If  the  strength  of  the  current  in  the  circuit  is  C,  the  magnetic 
force  at  P  is  directed  along  the  axis  of  the  circuit  (Fig.  65) 
and  is  numerically  equal  to  the  negative  of  the  derivative  with 
respect  to  x  of  Co>.  The  intensity  of  the  force  is,  therefore, 


and  at  the  centre  of  the  circuit,  where  x  =  0,  it  is  2-jrC  fa. 

This  result  evidently   agrees  with    the    awkward    statement 

sometimes  used  to  define  the  electromagnetic  unit  of  current. 

"  If  one  centimetre  of  a  linear  circuit 

which  carries  the  unit  current  be  bent 

into  an  arc  of  one  centimetre  radius, 

the  strength  of  the  field  at  the  centre 

of  the  arc,   due  to   this  portion   of 

the  circuit,  will  be  one  dyne."     The 

ampere,  which  is  the  practical  unit 

of  current  intensity,  is  one-tenth  of 

the  unit  just  defined. 

If  for  convenience  we  denote  the 
quantity    a/x    by    u    and    its    recip 

rocal  by  v,  the  potential  function  (Ceo)  just  found  may 
be  written  in  either  of  the  forms  27rC  Jl  —  I/  Vl  +  u*\  or 
2  TrCjl  —  v/Vl  +  v2J,  and,  according  as  x  is  greater  or  less 
than  a,  we  may  use  one  or  other  of  the  developments 


FIG.  66. 


2w2~2.4"4  +  2.4.6"6      '"\ 


If,  then,  PU  P2,  Ps,  •  •  •  represent  zonal  harmonics  expressed  in 
terms  of  cos  a,  and  if  ul  and  i\  represent  a/r  and  r/a,  the 


262  ELECTROMAGNETISM. 

value  of  the  potential  function  at  a  point  distant  r  from  the 
centre  of  the  circuit,  in  a  direction  (Fig.  66)  making  an  angle 
a  with  the  x  axis,  is  given  according  as  r  is  greater  or  less  than 
a  by  one  or  other  of  the  developments 


If  an  infinitely  long  straight  wire  which  carries  a  steady 
current,  (7,  forms  part  of  a  plane  closed  circuit,  all  the  other 
parts  of  which  are  at  infinity,  and  if  the  plane  of  the  circuit 
be  used  as  the  xz  plane  and  the  wire  as  the  z  axis,  the  solid 
angle  subtended  at  the  point  (x,  y,  z)  by  the  circuit  is 
2(7r  —  0),  where  tan  0  —  y/x.  The  force  components  at  the 
point  are,  then,  the  negatives  of  the  derivatives  with  respect 
to  x  and  y  respectively  of  2  C(ir  —  9  ),  that  is,  —  2  Cy/(x2  +  ?/2) 
and  H-  Cx  /  (x2  +  ?/2),  as  we  already  know. 

79.  The  Law  of  Laplace,  Mechanical  Action  on  a  Con 
ductor  which  carries  a  Current  in  a  Magnetic  Field.  It  will 
be  evident  from  the  discussion  on  page  218  that  the  strength 
of  the  magnetic  field,  H,  due  to  a  steady  current  of  C  electro 
magnetic  units  in  a  rigid  linear  circuit  may  also  be  computed, 
whatever  the  inductivity  of  the  homogeneous  surrounding 
medium,  on  the  assumption  that  every  element  ds  of  the  cir 
cuit  (Fig.  67)  makes  a  contribution  numerically  equal  to 

C  •  sin  (r,  ds)  •  ds  /r2, 

to  the  force  at  a  point  P,  where  r  is  the  distance  of  ds  from  P. 
The  direction  of  the  contribution  is  normal  to  the  plane  of  P 
and  ds,  and  such  that  a  north  magnetic  pole  at  P  tends  to 
whirl  right-handedly  about  a  straight  line  drawn  through  ds 
in  the  direction  of  the  current.  For  a  simple  illustration  of 


ELECTROMAGNETISM. 


263 


the  use  of  this  rule,  which  is  sometimes  called  "Laplace's 
Law,"  let  P  be  a  point  at  a  distance  r0  from  an  infinitely 
long  straight  wire  which  carries  a  current  C,  and  let  s  be 
the  distance  of  ds  from  the  foot  of  the  perpendicular  dropped 
from  P  upon  the  wire.  If  the  angle  (r,  ds)  be  denoted  by  6, 
s  =  r0  ctn  0,ds=  —  r0  esc2  6  d6,  r  =  r0  esc  0.  All  the  elements 
of  the  current  conspire  to  produce  at  P  a  magnetic  force  per 
pendicular  to  the  plane  of  P  and  the  wire.  The  magnitude 
of  this  force  is 


C 


£ 


*  sin  0  •  ds 


•  o  r< 

smOdO  =  —> 


as  before. 

If  a  circuit  is  not  plane,  the  different  elements  of  the 
current  will  contribute  to  the  magnetic  force,  at  a  point  P, 
elementary  forces  which  do  not  all 
have  the  same  directions.  In  this 
case  it  is  necessary  to  compute  sepa- 
rately  the  components  L,  M,  X  of  H. 
If  the  coordinates  of  the  beginning 
of  ds  are  Xj,  yl}  zl}  and  those  of  the 
end  #!  +  dxi,  y±  +  dy^  zl  -f-  dzlf  while 
those  of  P  are  or,  y,  z,  the  direction 
cosines  of  r  and  ds  are  (xl  —  x)/r, 

(!/i~y)/r>   Oi--)A>  and  dxi/ds, 

di/i/ds,  dz^/ds,  and,  if  the  direction 

cosines  of  dH,  the  contribution  to  the  force  at  P  made  by  the 

current  element  ds,  are  /,  m,  n,  then,  since  this  direction  is 

perpendicular  to  r  and  to  ds, 

l(xl  —  x)+  m(yi  -y)+n  (z^  -  z)  =  0, 

ldxv  +  mdyl  +  ndzl  =  0, 

I2  -h  m2  +  n2  =  1. 

If   we    represent   the    expressions    (yl  —  y)  dz±  —  (^  —  z)  dt/i, 
(zl  -  z)  dx1  -  (o-i  -  x)  dzly  fa  -  x)  dy^  -  (y±  -  y)  dxv  by  8',  8", 


FIG.  67. 


264  ELECTROMAGNETISM. 

8'"  respectively,  and  8'2  +  8"2  +  8'"2  by  8,  we  learn  from  these 
equations  that  I  =  8'/8,  m  =  8"/8,  n  =  8r"/8, 

cos  (>!,  ds)  =  [(a?!  -  a?)^  +  (yt  -  y)dy±  +  («!  -  z)dzl'\/rds) 

and  sin(r,  ds)  =  8/rds. 

If,  then,  the  components  of  dH  are  ^L,  dJf,  dN,  we  have 
the  equations  dL  =  (78'/r3,  <Of  =  <7S"/r3,  dN  =  C8"'/r3,  and 
from  these,  by  integration  over  the  circuit,  the  force  at  P  may 
be  computed. 

Since  action  and  reaction  are  equal  and  opposite,  a  unit 
magnetic  pole  at  P  would  exert  upon  the  element  ds  of  the 
conductor  which  carries  the  current  a  mechanical  or  "  pondero- 
motive"  force  the  components  of  which  would  be  —  CS'/r8, 
-  CS"/r8,  -  CB'"/r\  These  components,  written  in  terms  of 
the  components 

An  =  (*1  -  a)/**,     Mm  =  (2/1  -  y)l*>    ^m  =  («1  ~ 

of  the  magnetic  field  at  rfs  due  to  the  pole  at  P,  are 


and,  since  so  far  as  this  force  is  concerned  the  origin  of  the 
magnetic  field  is  immaterial,  these  expressions  give  the  com 
ponents  of  the  mechanical  force  which  act  upon  the  element 
ds  of  a  circuit  carrying  a  steady  current  C  in  any  magnetic 
field  which  at  ds  has  the  components  Lm,  Mm,  JVm. 

If  the  magnetic  field  at  dsl  —  an  element  of  a  linear  circuit 
sl  which  carries  a  steady  current  C\  —  is  due  to  a  steady  current 
C2  in  another  circuit  s2,  the  element  ds2  of  the  second  circuit 
at  the  point  (x2,  y^  z2)  contributes  to  the  magnetic  field  at  ds^ 
at  the  point  (x1}  y^  «x)  components  numerically  equal  to 


[202] 

^ 

-   [(2/1  -  2/2)  ^2  -  (#1  -  32 


ELECTROMAGNETISM.  265 

so  that  the  x  component  of  the  mechanical  force  exerted  upon 
the  circuit  element  ds±  by  the  circuit  element  dsz  is 

dXi  =  -^  \  [(?/!-  i/2)  dx,  -  (*!  -  *2)  dy*~\  dyl 

-  [(^1  -  *2)rf«2  -  (-1  - 
or      C*  (7  • 


i    zi»  _ 
or  2  -  -  [cos  (x,  r)  •  cos  (dsl}  ds2) 

-  cos  (x,  ds2)  cos  (r,  rfjjj  J  [203] 

where  r  is  the  distance  of  c?s2  from  rfs^ 

The  x  component,  Xlt  of  the  whole  mechanical  force  exerted 
upon  the  rigid  circuit  sl  by  the  rigid  circuit  s2  is  to  be  found 
by  integrating  the  expression  just  found  over  both  circuits. 

The  resulting  integral  will  evidently  not  be  changed  if  we 
add  to  the  integrand  any  quantity  which  disappears  when 
integrated  about  either  circuit,  and  this  fact  makes  it  possi 
ble  to  find  many  other  expressions  *  for  the  mechanical  force 
exerted  upon  an  element  of  one  circuit  by  an  element  of 
another,  which  will  account  mathematically  for  the  observed 
forces  between  two  rigid  closed  circuits. 

According  to  Ampere's  analysis,  the  resulting  action  between 
the  two  elements  dsl}  ds2  is  an  attraction  in  the  line  joining 
them  of  intensity 


*  For  exhaustive  treatments  of  this  important  subject  the  reader  should 
consult  Ampere,  Gilbert's  Ann.,  1821  ;  Ampere,  M6m.  de  V  Academic. 
1823,  1827;  TV.  Weber,  Ges.  Werke  ;  Grassmann,  Fogg.  Ann.,  1845; 
F.  E.  Neumann.  Abh.  BerL  Akademie,  1845  ;  TViedemann,  Lehre  von  der 
Elektricitdt  ;  Maxwell,  Treatise  on  Electricity  and  Magnetism,  §§  502-527  ; 
Webster,  Theory  of  Electricity  and  Ifagnetism,  §§  217-221.  For  conven 
ience  of  reference  I  have  followed  Professor  Webster's  order,  and  in  part 
his  notation  hi  the  brief  treatment  of  the  Electrodynamic  Potential  given 
in  Section  80.  See  Problem  307,  page  452. 


266  ELECTKOMAGNETISM. 

On  this  assumption  two  elements  in  the  same  straight  line 
repel  each  other  with  a  force  CiC2dslds2/r2,  while  two  parallel 
elements  perpendicular  to  the  line  which  joins  them  attract 
each  other  with  a  force  2  C^C^ds^ds^/r21.  These  expres 
sions,  like  those  which  precede,  hold  good  whether  the  ele 
ments  dsu  ds2  belong  to  the  same  circuit  or  to  two  different 
circuits. 

If  two  infinitely  long  straight  wires  (sj,  s2),  parallel  to  each 
other  at  a  distance  a  apart,  carry  in  the  same  direction  the 
steady  currents  Cl}  C2  respectively,  the  mechanical  force 

exerted  on  «!  by  s2  is  evidently  CiC2  1    I  [cos(x}r)/r2]ds1  -  ds2, 

J\  */2 

or  (2  OiC2/a)  (  dsi9  so  that  every  unit  length  of  sl  is  attracted 

towards  s2  with  a  force  of  2  CiC2/a  dynes. 

If  each  of  two  closed  circuits  (sl5  s2)  which  carry  steady  cur 
rents,  Ci,  <72,  consists  essentially  of  two  infinitely  long  wires 
parallel  to  the  z  axis,  if  the  currents  come  up  through  the  xy 
plane  in  the  two  circuits  at  the  points  (0,  a),  (c,  b)  respectively, 
and  go  down  at  the  points  (0,  —  a),  (c,  —  5),  the  first  circuit 
experiences  a  force  tending  to  urge  it  in  the  direction  of  the 
x  axis,  and  the  intensity  of  this  force  per  unit  length  of  both 
wires  of  8l  is  4 cdCi  {!/[(»  -  6)2+  c2]  -!/[(»  +  &)2+  c2] }. 


It  is  evident  from  the  discussion  of  the  properties  of  mag 
netic  shells  in  air  given  on  page  217  that  the  mechanical  action 
on  a  rigid  linear  circuit  carrying  a  steady  current  C  in  a 
magnetic  field  (caused  either  by  permanent  magnets  or  by  other 
currents  or  by  both)  may  be  mathematically  accounted  for  on 
the  supposition  that  every  element  ds  of  the  circuit  is  urged  by 
a  force  equal  to  C  ds  times  the  component  (F),  perpendicular 
to  ds,  of  the  total  magnetic  induction.  The  direction  D  of 
this  elementary  force  is  perpendicular  to  the  plane  of  C  and  F 
in  the  sense  shown  in  Fig.  68. 


ELECTROMAGNETISM.  267 

The  same  assumption  will  account  for  the  phenomena 
observed  when  a  deformable  circuit  is  placed  in  a  magnetic 
field. 

According  to  this  theory  the  component  in  any  direction  u 
of  the  force  on  the  element  ds  is  Cds  •  B  •  sin  (B,  ds)  cos  a,  where 
a  is  the  angle  between  n  and  the  normal  to 
the  plane  of  B  and  ds,  and  this  is  numeri 
cally  equal  to  the  volume  of  a  parallelepiped, 
adjacent  edges  of  which  are  represented  in 
magnitude  and  direction  by  Cds,  B,  and  a  unit 
length  in  the  direction  u.  This  volume  may 
also  be  represented  by  Cds  •  sm(uf  ds)  -B',  F 

where  B'  is  the  component  of  the  induction  B, 
normal  to  the  plane  of  u  and  ds,  and  this  expression  for  the 
force  component  is  occasionally  useful. 

If  (I,  m,  n)  are  the  direction  cosines  of  the  element  ds  and 
if  the  components  of  the  induction  B  are  Bx,  By,  BzJ 

sin  (B,  ds)  =  \  (m  •  B2  -  n  •  By) 2  +  (n  •  Bx  -  I  •  Bx) 2 
+  (I  •  By  -  m  •  B^\\/  \  B*  +  B*  +  B?\t 

and  the  resultant  electromagnetic  force  on  the  circuit  element 
ds  has  the  value 

C \  (m  •  Bz  -  n  •  Byy  +  (n  •  Bx  -  I •  Bs)*  +  (l-By-m.  Bx)2 \ I ,  ds. 

If  ds  is  an  element  of  a  current  filament  of  cross-section  o>  in  a 
massive  conductor  in  which  the  current  vector  is  q  or  (u,  v,  w), 
we  have  qu  =  C,  in*  =  1C,  vw  =  mC,  W<D  =  nC,  and  the  electro 
magnetic  force  may  be  written 

u\(v'Bt-w  Byf  +  (w  -Bx-u.  B;)2  +  (u  •  By  -  v  •  Bx)2^  ,  ds. 

The  components  parallel  to  the  coordinate  axes  of  the  electro 
magnetic  force  per  unit  volume  of  the  conductor  are,  therefore. 
(v-B2-iv-By),  (w-Bx-u.Bs),  (u.B,-v-Bx). 

If  the  element  ds  be  moved  parallel  to  itself  through  the 
distance  du,  the  mechanical  work  done  on  it  by  the  forces  of 


268  BLECTROMAGNETISM. 

the  field  can  be  represented  numerically  by  the  volume  of  a 
parallelepiped,  conterminous  edges  of  which  are  C ds,  B,  and 
du ;  this  volume  is  numerically  equal  to  C  times  the  number 
of  lines  of  induction  of  the  field  cut  by  the  element  during  the 
translation.  If  an  observer  be  imagined  to  lie  in  the  element 
in  such  a  way  that  the  current  enters  at  his  feet  and  goes  out 
at  his  head,  and  if  he  faces  in  such  a  direction  that  he  can 
look  along  the  lines  of  force,  the  work  done  by  the  translation 
will  be  positive  if  these  lines  appear  to  pass  him  from  left  to 
right,  that  is,  if  the  displacement  is  to  his  left.  It  is  easy  to 
see,  moreover,  that  if  the  element  ds  be  revolved  about  any 
axis  through  a  small  angle,  the  work  done  upon  it  may  be 
represented  by  C  times  the  number  of  lines  of  induction  cut 
by  the  element  during  the  displacement ;  we  may  infer,  there 
fore,  that  the  electromagnetic  work  done  by  the  field  upon  any 
portion  s  of  a  circuit  during  any  displacement  is  measured  by 
the  product  of  the  current  strength  and  the  number  of  lines 
of  induction  cut  by  s.  The  direction  in  which  a  rigid  closed 
linear  circuit  carrying  a  steady  current  C  in  a  magnetic  field 
of  any  kind  will  tend  to  move  may  be  inferred  from  the  fact 
that  the  circuit  will  behave  in  this  respect  like  the  equivalent 
magnetic  shell. 

It  is  easy  to  see  from  the  discussion  on  page  216  that  the 
mutual  potential  energy  of  an  external  field  and  the  mag 
netic  shell  mechanically  equivalent  to  a  given  circuit,  —  that 
is,  the  mechanical  work  that  must  be  done  to  bring  the  shell 
already  formed  into  the  field,  —  is  equal  to  —  CN,  where  N 
is  the  whole  number  of  lines  (unit  tubes)  of  induction  of  the 
field  which  the  current  surrounds  right-handedly.  The  cir 
cuit  will  tend  to  move,  therefore,  so  as  to  make  N  as  large  as 
possible.  If,  for  instance,  a  plane  circuit  of  area  A  carries  a 
steady  current  C  in  a  uniform  field  of  induction  of  intensity 
By  any  motion  of  the  circuit  parallel  to  itself  would  not 
change  the  induction  through  it,  and  there  is  no  tendency  to 
any  such  motion;  if  the  normal  to  the  plane  of  the  circuit 
makes  an  angle  0  with  the  direction  of  the  field,  a  couple, 


ELECTKOMAGNETISM.  269 

of  moment  C  •  A  •  B  •  sin  0,  acts  on  the  circuit  and  tends  to 
decrease  6. 

If  into  a  magnetic  field  FQ  which  has  the  components 
XQJ  F0,  ZQ  a  linear  circuit  carrying  a  steady  current  be  intro 
duced,  and  if  the  electromagnetic  field  due  to  the  current  alone  is 
FU  or  (JCi,  Pi,  ZJ,  the  whole  field  is  (JT0  +  A'1?  F0  +  F1?  ^0  4-  3i), 
and  the  whole  magnetic  energy  in  the  field  is 


r 


or  ^          + 


+  Fori  + 


The  first  integral  is  the  magnetic  energy  of  the  original 
field,  the  second  that  of  the  field  of  the  circuit  alone,  and  the 
third  the  magnetic  energy  due  to  the  introduction  of  the  circuit 
when  formed  into  the  field.  "We  may  now  show  that  this  last 
term,  which  may  be  written 


cos 


is  equal  to  the  product  of  the  strength  of  the  current  and 
the  flux  of  induction  of  the  original  field  in  the  positive  direc 
tion  through  the  circuit.  Since  all  the  equipotential  surfaces 
of  the  field  Fl  are  bounded  by  the  circuit,  we  may  cap  the 
circuit  by  a  whole  series  *  of  such  surfaces  and  write  the 

*  A.  Gray,  Treatise  on  Magnetism  and  Electricity,  Vol.  I,  p.  293. 


270  ELECTKOMAGNET1SM. 

total  induction  through  the  circuit  due  to  the  outside  field  in 
the  form 


M  =  (IX,  +  m  ro  +  nZ0)  dS  =  ^o  •  cos  (F0,  F,)  dS, 

where  the  integration  is  to  be  taken  over  any  one  of  these 
caps  and  where  /,  m,  n  are  the  direction  cosines  of  the  normal 
to  the  cap. 

If  a  unit  magnetic  pole  were  carried  around  any  line  of  force 
Sj  of  the  field  Fl}  the  work  done  on  it  would  be  4?r  times 

the  current  C  in  the  circuit,  so  that  4  irC  =  I  F^-  ds±.  If  we 
multiply  each  side  of  this  last  equation  by  M,  we  have 


C  CC  p(lX9  +  mY0 


cos 


Since  the  caps  are  equipotential,  Fl  ds±  has  the  same  value  for 
all  lines  of  force  between  any  two  caps,  and  since  the  induction 
JJ,FQ  is  solenoidal,  the  first  integral  factor  of  the  second  mem 
ber  has  the  same  value  for  all  the  caps.  We  may  find  the 
value  of  the  second  member,  therefore,  by  imagining  space 
divided  up  into  elements  which  are  portions  of  tubes  of  force 
of  the  field  F±  bounded  by  equipotential  surfaces  of  this  field, 
multiplying  the  volume  of  each  element  by  the  value  in  it  of 
/J.FQ  -  FI  •  cos  (F0,  F-L),  and  finding  the  limit  of  the  sum  of  all 
these  quantities  divided  by  47r.  The  value  of  the  volume 
integral  must  be,  however,  independent  of  the  shapes  of  the 
elements,  and  we  have,  in  general, 


cos  (F0,  F$  dr, 


ELECTROMAGNETISM.  271 

The  magnetic  energy  in  the  medium  is  often  called  the  "elec- 
trokinetic  energy."  That  portion  of  the  electrokinetic  energy 
which  is  due  to  the  introduction  of  the  circuit  already  established 
into  the  given  field  is  evidently  the  negative  of  the  mutual  poten 
tial  energy,  corresponding  to  work  done  against  mechanical 
forces,  of  the  equivalent  magnetic  shell  and  the  field. 

If  a  portion  s  of  a  circuit  electrically  connected  through 
mercury  cups  with  the  rest  of  the  circuit,  which  is  fixed,  be 
rotated  and  finally  brought  back  to  its  original  position*,  elec 
tromagnetic  work  will  be  done  on  s  if  it  cut  lines  of  the  field  in 
positive  direction  during  the  motion,  but  the  whole  circuit  may 
be  represented  by  . 

the  same  magnetic  Q — — — L«* -L 

shell  at  the  begin-  _-/ 

ning  and  at  the  end  -=F- 

of  the  process,  and          -Er 
the   mutual  poten-       -p 

tial  energy  of  the   p  /  ^ ^ 

circuit    and    the 

field  is  unaltered  by  FIG.  09. 

the  displacement. 

Under  these  circumstances,  as  will  appear  in  the  sequel,  cur 
rents  are  induced  in  s  by  the  motion. 

If  in  the  case  of  the  circuit  shown  in  Fig.  69  the  conductor 
AB  is  free  to  slide  on  the  rails  DA,  GB  in  such  a  way  as  to  be 
always  parallel  to  DG,  it  will  move  in  the  direction  indicated 
by  the  detached  arrow,  the  circuit  will  be  made  to  embrace  in 
the  positive  direction  a  greater  number  of  lines  of  induction, 
and  the  electrokinetic  energy  will  be  increased.  If  the  motion 
take  place  without  external  help,  the  necessary  energy  must  be 
furnished  at  the  expense  of  chemical  action  in  the  battery. 
Let  E  be  the  electromotive  force  of  the  battery,  r  the  resistance 
of  the  circuit  at  any  instant,  and  C  the  current  which  then 
passes  through  it :  the  energy  furnished  by  the  chemical  action 
in  the  battery  during  the  time  dt  will  be  ECdt,  and  of  this  a 


272 


ELECTROMAGNETISM. 


/A 

FIG.  70. 


part,  at  least,  C2rdt,  appears  as  heat  in  the  conductors  which 
make  up  the  circuit.  If  AB  be  held  still,  C  will  have  such  a 
value,  CQ,  that  ECQ  =  C0*r.  If,  however,  AB  be  moving  toward 
the  right,  the  current  will  be  smaller  than  COJ  EC  will  be 

a  fraction  of  ECQ9 
B/  _  C2r  a  smaller  frac- 
tion  of  C0V,  and 
EC  will,  therefore, 
be  greater  than 
C2r.  The  difference 
(EC  -  C*r)  dt  now 
represents  the  work 
done  during  the 
time  dt  in  moving 

AB  :  a  part  of  this  work  is  used  in  overcoming  friction  on  the 
rails,  a  part  in  communicating  kinetic  energy  to  AB,  and  a 
third  part  in  increasing  the  energy  of  the  medium.  If  for 
convenience  we  denote  (EC  —  (7V)  dt  by  C  •  dp,  we  shall  have 
E  —  Dtp  =  Cr,  and  the  current  is  the  same  as  if  there  were 
in  the  circuit  an  electromotive  force  Dtp  opposed  to  that  of 
the  battery.  If  an  external  force  were  applied  to  AB  tending 
to  move  it  to  the  right,  the  velocity  might  be  increased  so  much 
that  the  current  would  be  reduced  to  zero 
or  caused  to  flow  in  the  opposite  direction. 
If,  however,  AB  were  forced  to  move  to 
the  left  by  external  forces,  the  current  in 
the  circuit  would  become  greater  than  C0 
and  would  have  the  same  direction  as  E. 
Fig.  70  illustrates  a  case  where  the 
resultant  magnetic  field  is,  as  before, 
normal  to  the  plane  of  the  circuit,  though 
the  field  lines  thread  the  circuit  in  the 

negative  direction  ;  in  this  case  AB  will  tend  to  move  toward 
the  left. 

Fig.   71    represents    Faraday's   metal  disc,  mounted    on  a 
metallic  arbor  and  free  to  turn  about  a  horizontal  axis.     At 


FIG.  71. 


ELECTROMAGNETISM. 


273 


any  instant  the  current  flows  in  the  disc  from  the  centre  to 

the  brush  P  and  the  conductor  which  carries  the  current  is 

urged  to  turn  in  the  direction  indicated  by  the  arrow.     The 

energy  in  the  medium  is  not  increased  by  the  motion  of  the 

disc,  and  the  work  done  by  the  battery  is  spent  in  heating 

the  conductors  in  the  circuit,  in  overcoming  friction  and  the 

resistance  of  the  air,  and  in  increasing  the 

kinetic  energy  of  the  disc.     If  the  field  is 

uniform,  if  S  is  the  area  of  one  face  of  the 

disc,  and  if  the  media  are  of  unit  induc- 

tivity,  the  work  done  on  the  disc  each  turn 

is  CHS,  and  if  it  is  making  n  turns  per 

second,  we  have   EC  =  C*r  +  CHSn.     If 

the  disc  be  used  as  a  motor  to  overcome 

resistance  of  some  kind,  and  if  the  energy 

required  per  turn  is/(?*),  CHS  =f(?i),  and 

from  these  two  equations  n  and  C  may  be 

found,  if  f  be  a  known  function. 

In  the  arrangement  shown  in  Fig.  72 
a  rigid  wire  free  to  turn  about  the  axis  of 
a  fixed  vertical  magnet  makes  electrical 
contact  with  the  magnet  at  its  middle. 
The  current  from  a  battery  flows  through 
a  circuit  made  up  of  the  wire,  the  mag 
net,  and  a  supplementary  fixed  conductor 
forming  a  prolongation  of  the  axis  of  the 
magnet.  In  this  case  the  wire  will  turn  continuously  in  the 
direction  indicated.  It  is  easy  to  show  that  a  fixed  magnetic 
field  cannot  cause  continued  rotation  of  a  complete  rigid  circuit 
about  a  fixed  axis. 

80.  The  Electrodynamic  Potential.  If  while  a  linear  cir 
cuit,  s2,  which  carries  a  steady  current,  CZ)  remains  fixed,  a 
neighboring  linear  circuit,  s^  which  carries  a  steady  current, 
<?!,  is  deformed  or  moved  without  being  stretched,  so  that 
every  element  dsl  is  unchanged  in  length  but  the  coordinates 


FIG.  72. 


274  ELECTROMAGNETISM. 

of  the  beginning  of  the  element  receive  increments  Sxl}  8ylt  Bzl} 
which  are  analytic  functions  of  xly  yly  z1}  the  work  done  by  the 
forces  which  s2  exerts  upon  sl  is  approximately  equal  to 


dZi 
or  to 


2  -f- 


[dx2  •  Sec!  -f  ^2/2  •  8i/i  +  c?«2  •  8«i]- 

The  first  factor  under  the  integral  signs  in  the  second 
integral  of  the  last  expression  is  equal  to  J)  (l/r)  -ds^  and 
if  we  integrate  the  whole  integrand  by  parts  with  respect  to  s1? 
we  get 

[(da,  •  Sa?!  +  dy*  •  8^  +  d«2  •  8«i)  /r]  taken  between  limits 


where  the  expression  in  brackets,  having  the  same  value  at 
both  limits,  can  be  omitted.  The  expression  for  the  elemen 
tary  work  done  on  sl  during  its  displacement  is,  therefore, 


-L  •  dx2  +  dyi  •  dyz  +  dzl  •  dzz~\ 
+  CiC2  f  C(dx2  •  d8Xl  +  dyz  -  dfyi  +  dzi  •  d&zj  /r, 

c/l*/2 

and  this  is  obviously  equal  to  the  variation  of  the  integral 
Ci  C,  f  C(dXl  •  dx.  +  dyl  •  dy*  +  dz,.  -  dz2)  /r, 

*/l«/2 


ELECTROMAGNETISM.  275 

caused  by  the  elementary  displacement.     This   last  integral 
written  in  the  form 


gives  what  is  often  called  F.  E.  Neumann's  Expression  for 
the  Elect  rodynamic  Potential.  The  increase  in  the  value  of 
this  function  caused  by  any  finite  displacement  of  s^  evidently 
represents  the  work  done  on  sl  by  the  field  due  to  s2  during 
the  displacement:  this  work  depends  only  upon  the  original 
and  final  configurations.  The  Electrodynarnic  Potential  corre 
sponds  to  that  portion  of  the  electrokinetic  energy  which  is  due 
to  the  mutual  proximity  of  the  circuits.  Its  negative  is  equal 
to  what  is  sometimes  called  the  mutual  potential  energy  due  to 
the  mechanical  forces  acting  between  the  circuits.  It  is  impor 
tant  to  notice  that  although  the  ponderomotive  forces  which 
urge  a  rigid  circuit  carrying  a  given  current,  (7,  in  a  magnetic 
field  can  be  correctly  found  from  the  expression  for  the  mutual 
potential  energy  of  the  field  and  a  magnetic  shell  of  strength 
C  bounded  by  the  circuit,  this  may  be  regarded  from  one 
point  of  view  as  merely  a  convenient  mathematical  device.  If 
the  shell  were  to  move  under  the  action  of  the  field  alone 
and  acquire  kinetic  energy  and  overcome  external  resistance, 
this  work  would  be  done  at  the  expense  of  the  mutual  poten 
tial  energy  of  the  field  and  the  shell.  If,  C  being  kept 
constant,  the  circuit  were  to  move  under  the  action  of  the 
field  in  exactly  the  same  way,  the  work  would  be  done  at  the 
expense  of  the  generator  which  maintains  the  current.  In 
other  words,  there  is  no  sensible  mutual  potential  energy  of 
the  field  and  the  circuit,  the  exhaustion  of  which  measures  the 
work  done  by  the  forces  of  the  field  during  any  displacement 
of  the  circuit. 

The  integrand  in  the  expression  given  by  Neumann  can  be 
increased  at  pleasure  by  any  quantity  which  disappears  when 
integrated  around  either  sx  or  s.2.  Such  a  quantity  is  X  •  DgiD^i', 
or  A[cos(r,  ds^  -cos(r,  dsz)  —  cos(dsl}  ds^/r,  where  A  is  any 


276 


BLBCTROMAGNETISM. 


constant.     The   corresponding   form  of   the   Electrodynamic 
Potential  is 

Ci C2  \    \  \\  cos (r,  ds-t)  •  cos (r,  ds2)  \(l/r) ds^  -  ds^ 

i  -  x)  • cos  (ds»  «&»)  •  C1  /r)  \  dsi  •  dsr 

A  form  sometimes  convenient  is  obtained  by  putting  X  =  1. 

In  the  case  of  two  vertical,  coaxial,  circular  wire  circuits 
of  radii  rx  and  rz,  at  a  distance  a  apart  (Fig.  73),  we  may 


denote  by  fa  and  fa  the  angles  which  radii,  drawn  from  dslt 
ds2  respectively  to  the  centres  of  their  circuits,  make  with  the 
vertical  and  put  xt  —  TI  cos  fa,  x2  =  r2  cos  fa,  yl  —  r±  sin  fa, 
yz  =  r2  sin  fa,  r*  =  a*  +  r,2  +  r22  -  2  r^  •  cos  (</>  -  ^a).  The 
expression 

P  =  d  <72  f  f(^!  •  ^x2  +  %!  •  ^2/2  -h  dzl  •  dz2)  /T 

then  becomes 

cos(fa-fa)dfa 


or 


C^Qdfa. 

*/o 


ELECTROMAGXETISM.  277 

That  the  definite  integral  Q  is  not  a  function  of  <f>2  follows 
from  the  fact  that  the  definite  integral  which  represents  its 
partial  derivative  with  respect  to  <£2  is  the  limit  of  the  sum 
of  elements  which  destroy  each  other  in  pairs  :  we  may  there 
fore  give  to  <£2  in  the  expression  for  Q  any  convenient  value 
(say  zero)  and  write  P  —  2-n-  C^r^Q.  We  may  conveniently 
transform  the  integral  which  represents  Q  by  putting 

2  0    = 

and  get 


or  »  -  -\  E\, 

where  K  and  E  are  the  complete  elliptic  integrals  of  the  first 
and  second  kinds.  The  numerical  values  of  these  integrals 
for  various  values  of  A*  are  to  be  found  in  "  A  Short  Table  of 
Integrals  "  (Ginn  &  Company,  Boston).  It  is  to  be  noted  that 
if  in  this  analysis  we  imagine  finite  currents  to  be  carried 
by  conductors  of  zero  cross-section,  and  -t\  and  r2  to  be  equal, 
then,  if  a  approaches  zero,  A-  approaches  unity  and  P  grows 
large  without  limit.  The  derivative  of  P  with  respect  to  a 
gives  in  general  the  mutual  attraction  of  the  two  circuits. 

If  the  external  field  about  a  linear  circuit  s1}  carrying  a 
circuit  CD  is  due  to  a  current  C2  in  another  linear  circuit  s2, 
we  have  two  different  expressions  for  the  mutual  potential 
energy  of  the  magnetic  shells  which  correspond  to  the  two 
circuits.  These  are  —  C^Y,  where  N  is  the  number  of  lines 
of  induction  due  to  C2  which  thread  sl  positively,  and  the 
negative  of  the  Electrodynamic  Potential  of  the  two  circuits. 
When  Ci  and  C.2  are  both  unity  the  Electrodynamic  Potential 
measures  the  magnetic  induction  through  either  circuit  when 
the  unit  current  traverses  the  other. 

The  number  of  lines  of  magnetic  induction  which  thread 
either  of  two  simple  linear  circuits,  made  of  non-magnetic 
material  and  removed  from  the  neighborhood  of  other  currents 


278  ELECTROMAGNETISM. 

and  permanent  magnets,  when  the  unit  current  passes  through 
the  other  circuit,  is  called  the  coefficient  of  mutual  induction 
or  the  mutual  inductance  of  the  two  circuits.  The  numerical 
value  of  this  coefficient  depends  upon  the  character  of  the 
media  in  the  neighborhood  of  the  circuit. 

If  two  exactly  similar  linear  circuits,  SL  and  s2,  carrying 
steady  currents  of  unit  intensity,  lie  side  by  side,  and  if  one  of 
them  (.92)  be  imagined  to  move  up  towards  coincidence  with 
the  other,  the  value  of  the  integral  which  represents  the  Elec- 
trodynamic  Potential  approaches  the  form 

C  CGOS  (dsn  ds2)  ds-L  -  ds2 
Li  =  I    I ) 


where  the  integration  is  to  be  extended  twice  over  the  same 
circuit.  If  the  circuits  are  supposed  to  be  mere  geometrical 
lines,  the  value  of  this  integral  will  be  in  general  infinite ; 
if,  however,  sx  and  s2  are  made  of  wires  of  small  but  definite 
cross-sections,  the  finite  limit,  as  s2  is  moved  into  close  contact 
with  sly  of  the  flux  of  magnetic  induction  caused  by  the  unit 
current  in  s2  through  a  diaphragm  bounded  by  sl  is  practically 
the  flux  through  the  diaphragm  due  to  the  unit  current  in  s^ 

The  number  of  lines  (unit  tubes)  of  magnetic  induction 
which  thread  a  simple  fine  wire  circuit  made  of  non-magnetic 
material,  which  carries  a  steady  current  of  unit  strength  when 
there  are  no  other  currents  and  no  permanent  magnets  in 
its  neighborhood,  is  very  nearly  equal  to  what  is  called  the 
coefficient  of  self-induction  or  the  self -inductance  of  the  simple 
circuit,  under  the  circumstances.  The  numerical  value  of  this 
coefficient,  which  we  shall  soon  be  able  to  define  more  accu 
rately,  depends  very  much  upon  the  nature  of  the  media  about 
the  circuit. 

81.  Coefficients  of  Induction.  If  two  fine  wire  closed  cir 
cuits  of  non-magnetic  material,  exactly  alike  in  size  and  shape, 
and  carrying  in  the  same  direction  steady  currents  of  intensity 
C'  and  C"  respectively,  are  placed  as  nearly  as  possible  in 


BLECTROMAGNETISM.  270 

coincidence,  the  coefficient  of  mutual  induction  of  the  two  is 
practically  the  same  as  the  coefficient  of  self-induction  (L)  of 
either,  and  the  work  required  to  separate  the  two  circuits  to 
an  infinite  distance  from  each  other  is  C'C"L.     If,  then,  a 
fine  wire  closed  circuit  which  carries  a  steady  current  C  be 
supposed  made  up  of  infinitely  slender  closed  circuit  filaments 
lying  freely  in  contact,  it  is  easy  to  get  an  expression  for  the 
work  that  must  be  done  in  removing  these  filaments  one  after 
another  out  of  the  field.     If  at  some  stage  in  the  process  the 
remaining  filaments  carry  altogether  the  current  C  —  C",  the 
work  required  to  remove  another  filament  carrying  the  cur 
rent  dC"  would  be  (C-  C")dC"-L,  and  this  integrated  with 
respect  to   C"  between  0  and   C  yields  %  C2L,  which  is  an 
expression  for  the  intrinsic  energy  of  the  original  collection  of 
filaments.     Again,  if  a  current  C  be  set  up  and  kept  steady 
in  any  closed  circuit  in  a  medium  of  any  kind  which  contains 
no  permanent  magnets  and  no  other  currents,  the  medium 
becomes  polarized  by  induction  and  is  a  field  of  force.     The 
electrokinetic  energy  is  equal  to  the  volume  integral  taken 
over  all  space  of  /x(72J?2/8  TT,  where  R  is  the  intensity  of  the 
field  due  to  a  unit  current  in  the  conductor.     It  is  easy  to  see 
that  this  reduces  in  the  case  of  a  linear  circuit  to  %  C2  times 
what  we  have  called  the  coefficient  of  self-induction  of  the 
circuit,  and  we  are  led  to  define  the  coefficient  of  self-induction 
of  a  circuit,  made  up  of  conductors  of  any  form  surrounded  by 
media  the  susceptibilities  of  which  are  independent  at  every 
point  of  the  intensity  of  the  force  at  the  point,  as  twice  the 
energy  in  the  magnetic  field  when  the  circuit  carries  a  current 
of  one  electromagnetic  unit  and  there  are  no  other  currents 
and  no  permanent  magnets  in  the  neighborhood. 

If,  for  instance,  a  uniformly  distributed  current  C  be  carried 
lengthwise  in  a  homogeneous,  infinitely  long  cylinder  of  revolu 
tion,  of  radius  a,  and  be  brought  back  in  a  thin  cylindrical  shell 
of  inside  radius  b  and  outside  radius  c,  coaxial  with  the  cylin 
der,  there  is  no  field  without  the  shell ;  the  intensity  of  the 
field  is  2  Cr/a*  within  the  cylinder,  2  C/r  between  the  cylinder 


280  ELECTROMAGNETISM. 

and  the  shell,  and  2  C(c2  —  r*)/r  (c2  —  b2)  in  the  shell  itself. 
Neglecting  the  space  occupied  by  the  thin  shell,  which  would 
contribute  little  to  the  result,  the  whole  energy  in  the  field  per 
unit  length  of  the  cylinder  is 


^  f 

a*    Jo 


2  i«*dr  +      -  4  C2         vr  •  dr. 

8  7T 


If  the  medium  between  the  shell  and  the  cylinder  has  the 
uniform  inductivity  /x,2,  this  energy  is  i/xjC2  +  n2CzIogb/a. 
The  coefficient  of  self-induction  of  the  circuit  per  unit  length 
is,  therefore,  when  the  shell  is  thin,  \^  +  2  p2logb/a. 

The  coefficient  of  self-induction,  in  electromagnetic  absolute 
c.g.s.  units,  of  a  circular  ring  of  circumference  I,  made  of  non 
magnetic  wire  of  radius  r  and  surrounded  by  air,  is,  according 
to  Kirchhoff,  2  I  [log  (l/r)  —  1.508],  and  that  of  a  square  circuit 
of  perimeter  I,  made  of  similar  wire,  2  I  [log  (l/r)  —  1.910]. 
Regarding  the  coefficient  of  self-induction  from  the  point  of 
view  of  the  energy  in  the  field,  it  is  possible  to  prove  that  the 
coefficient  of  a  part  of  a  circuit  consisting  of  a  straight  wire  of 
length  I  and  radius  r  is  approximately  2  I  [log  (2  Z/r)  +  i  /*  —  1], 
where  /x  is  the  magnetic  permeability  of  the  wire.  For  addi 
tional  examples,  the  reader  is  referred  to  Winkelmann's  Hand- 
buck  der  Physikj  Vol.  Ill,  Maxwell's  Treatise  on  Electricity 
and  Magnetism,  Vol.  II,  and  to  Gray's  Absolute  Measurements 
in  Electricity  and  Magnetism,  Vol.  II. 

If  Xlf  YI,  Zi  are  the  components  of  the  electromagnetic  field 
which  a  unit  current  flowing  in  a  given  circuit  sx  of  self-induc 
tance  L1  would  cause  if  the  surrounding  space  contained  no 
other  currents  and  no  permanent  magnets,  and  if  this  space  is 
already  the  seat  of  a  magnetic  field  X,  Y,  Z,  caused  either  by 
currents  or  by  permanent  magnets,  or  by  both,  then  if  a  steady 
current  Ci  be  set  up  and  maintained  in  slt  the  electrokinetic 
energy  is 


<^f  ff  1*1(0^  +  xy 


ELECTROMAGNETISM.  281 

The  integrand  can  be  split  up  into  three  terms, 


and  2nd  \X±X  4  Yl  Y  4  Z^Z], 

and  the  corresponding  integrals  represent  respectively  -J-  C-fL-^ 
the  energy  of  the  original  field,  and  that  part  of  the  electro- 
kinetic  energy  due  to  the  introduction  of  the  current  into  the 
field.  If  the  external  field  is  due  to  a  steady  current  C2  in 
a  second  circuit  s2  of  self-inductance  L2)  the  second  integral  is 
•J-  C^LK  and  if  the  third  be  written  dd^f,  the  whole  energy 
becomes  £  CfL^  4  Mdd  4  i  CJL*  The  quantity  M,  which 
in  the  case  where  s^  and  s2  are  linear  is  the  coefficient  of  mutual 
induction  of  the  two  circuits,  serves  to  define  this  coefficient 
in  the  case  of  circuits  which  are  not  linear,  surrounded  by 
media  which  have  susceptibilities  independent  of  the  strength 
of  the  field. 

If  n  circuits  which  have  self-inductances  Llf  L2,  LS)  •  •  •  and 
carry  currents  C1?  C2,  C3,  •  •  •  exist  together  in  a  soft  medium, 
and  if  the  mutual  inductance  of  the  pth  and  kth  circuits  is  Mpk, 
the  elect  rokinetic  energy  T  is  equal  to 

f  4-  L2d*  4-  L8d*  4-  •  •  •  4-  LnC^ 
4  MdC  +  MdC  4  •  •  •  4  Mlnd 


where  the  values  of  the  inductances  depend  upon  the  configura 
tion  of  the  system.  If  this  configuration  is  determined  by 
a  number  of  generalized  coordinates  qv  q2,  q^  •  •  •,  the  electro- 
dynamic  force,  in  the  Lagrangian  sense,  which  tends  to  increase 
any  one  of  these  coordinates  (leaving  the  rest  unchanged)  is 
the  partial  derivative  of  T  with  respect  to  this  coordinate. 
If  every  circuit  is  rigid,  the  L's  are  constant  during  any  change 
of  configuration. 

82.  Maxwell's  Current  Equations.  Various  Current  Sys 
tems.  We  may  infer  from  experiment  that  if  a  unit  magnetic 
pole  be  moved  about  a  simple  closed  path  in  any  steady  electro 
magnetic  field,  whether  the  medium  in  which  the  part  lies  is 


282  ELECTHOMAGNETISM. 

homogeneous  or  not,  the  work  done  on  it  by  the  field  is  equal 
to  47r<7,  where  C  is  the  whole  current  which  passes  in  positive 
direction  through  any  surface  or  diaphragm  which  caps  the 
path.  If  u,  v,  w  are  the  components  of  the  current  intensity, 
the  flux  through  the  cap  may  be  written  in  the  form 

J  J  [u-  cos  (x,  ri)+v>  cos  (y,  n)  +  w>  cos  (z,  n)']  dS, 

and  if  L,  M,  N  are  the  components  of  the  magnetic  force 
H,  the  line  integral  of  H  taken  around  the  path  is  equal, 
according  to  Stokes's  Theorem,  to 


~  D*M)  •  cos  (x>  n)  +  (P*L  ~  D*N)  •  cos  (y,  n) 
+  (DXM  -  DyL)  -  cos  («, 
It  follows  that  the  integral 


Jj  [(4 


-  DyN  +  DZM)  cos  (x,  ri) 

+  (4  TTV  -  DZL  -f  DXN)  cos  (y,  n) 
+  4-7rw-  DM  +  Z)L  cos  z 


must  vanish  whatever  the  shape  of  the  cap  and,  therefore, 
that  at  every  point 

47m  = 


=DZL-DXN, 

=  DXM  -  DyL.  [205] 

These  are  Maxwell's  Current  Equations,  which  can  be  stated 
in  the  single  vector  equation 

4  Tn/  =  curl  of  H. 

This  has  been  called  by  Heaviside  "  the  first  circuital  equa 
tion"  of  the  electromagnetic  field.  It  states  that  4  TT  times 
the  resolved  part  of  the  current  intensity  at  any  point  within 
a  conductor,  in  any  direction,  is  equal  to  the  resolved  part  in 
the  given  direction  of  the  curl  of  the  magnetic  force.  The 
equation  holds  even  in  a  non-homogeneous  medium. 


ELECTROMAGNETISM.  283 

Maxwell's  Equations,  with  the  characteristic  volume  and 
boundary  differential  equations  which  the  magnetic  induction, 
as  we  have  seen,  must  always  satisfy,  completely  determine  a 
steady  magnetic  field  in  given  media,  when  the  cm-rent  q  is 
known. 

In  any  homogeneous  soft  medium  the  magnetic  force  H  is 
solenoidal,  and  we  may  infer  from  the  work  of  Section  69 
that  it  has  a  vector  potential  function  Q  equal  to  Pot  q.  We 
have,  therefore,  H  =  curl  Q,  4  irq  =  curl  H,  and,  if  the  compo 
nents  of  H  and  Q  are  L,  M,  N  and  Qx,  Qy,  Qz  respectively, 


L  =  DvQz-DzQv     M=DZQX-DXQZ,     N=DxQy-DyQ,. 

When  in  a  steady  field  H  is  known,  Maxwell's  Equations, 
or  their  equivalent,  give  the  current  vector  q  directly.  If,  for 
example,  the  magnetic  force  is  zero  everywhere  without  an 
infinitely  long  cylindrical  surface  A  B 

S  of  any  shape,  while  within  S  the   K  —  I  I  -  K* 

field  has  the  uniform  strength  N,  F      "4       ° 

and  is  directed  parallel  to  the  gen 

erating  lines  of  the  surface,  q  is  zero  within  and  without  S. 
To  show  that  S  itself  is  a  current  surface,  let  KK  be  a  por 
tion  of  a  generating  line  drawn  in  the  direction  of  the  field 
within,  and  let  AB  and  CD  be  lines  each  of  length  I  parallel 
and  close  to  KK1,  one  within  and  the  other  without  S,  drawn 
so  that  AC  and  BD  are  normal  to  the  surface.  The  line  inte 
gral  of  the  magnetic  force  taken  around  the  perimeter  of 
the  rectangle  ACDB  is  numerically  equal  to  IN,  so  that,  by 
Stokes's  Theorem,  the  surface  integral  of  the  normal  upward 
component  of  the  curl  taken  over  the  area  of  the  rectangle  is 
IN,  and  this  is  equal  to  4  TT  times  the  steady  flow  of  electricity 
through  the  rectangle.  There  is,  therefore,  a  uniform  flow  of 
electricity  in  S  perpendicular  to  its  generating  lines  equal  to 
N/4:  TT  per  unit  length  of  the  surface. 


284  ELECTEOMAGNETISM. 

This  is  practically  the  case  of  an  electromagnetic  solenoid, 
that  is,  an  infinitely  long  cylindrical  surface  wound  uniformly 
(and  as  nearly  perpendicularly  to  the  axis  of  the  cylinder  as 
possible)  with  turns  of  fine  wire.  If  there  are  n  turns  on  each 
centimetre  of  length  of  the  cylinder  (Fig.  75)  and  if  each 
turn  carries  a  steady  current  (7,  N  /kir  =  nC,  or  N  =•  kirnC. 

This  result  is  independent  of  the  magnetic  inductivity  of 
the  homogeneous  soft  medium  within  the  cylinder.  The 
induction  in  the  medium  is  47m/A(7,  and  the  intensity  of  its 
polarization  (magnetization)  is  kirnkC  or  nC(p  —  1).  The 
coefficient  of  self-induction  per  unit  length  of  the  solenoid 
is  4  Trn^^A,  where  A  is  the  area  of  the  cross-section  of  the 
cylinder. 

If  a  part  of  the  space  within  the  solenoid  be  taken  up  with 
a  homogeneous  soft  medium  of  permeability  pi,  and  the  rest 

by  an  infinitely  long 
cylinder  of  another 
FIG  75  homogeneous  soft 

medium  of  permea 

bility  /z2,  the  lines  of  which  are  parallel  to  those  of  the  sur 
face  upon  which  the  wire  is  wound,  the  lines  of  force  are 
unchanged  in  form,  the  induction  in  the  first  medium  is  4  -n-n^  C 
and  in  the  other  4  -rrn^  C.  If  Al  and  A2  represent  the  portions 
of  the  cross-section  A  of  the  solenoid  occupied  by  the  two 
media,  the  self-inductance  of  the  solenoid  per  unit  length  is 


The  coefficient  of  mutual  induction  of  two  infinitely  long 
solenoids  S-^  S2,  one  of  which  has  n^  turns  and  the  other  nz 
turns  per  unit  of  its  length,  is  zero,  unless  one,  say  $2,  is  within 
the  other.  In  this  case  the  coefficient  has  the  value  4  irn^n^A^ 
per  unit  length  of  the  two,  where  A2  is  the  area  of  the  cross- 
section  of  $2- 

If  two  infinitely  long,  cylindrical  surfaces,  whatever  their 
shapes  may  be,  have  parallel  generating  lines,  and  if  one  of 


ELECTROMAGNETISM.  285 

these  surfaces  is  within  the  other,  the  space  between  the 
surfaces  will  be  a  uniform  field  of  magnetic  force  of  strength 
Nj  directed  parallel  to  the  generating  lines,  and  the  regions 
without  the  outer  surface  and  within  the  inner  one  will  be 
fields  of  no  force,  if  a  uniform  current  of  strength  N/±TT  per 
unit  length  flows  in  each  surface  perpendicular  to  the  gen 
erating  lines  and  if  the  directions  of  flow  around  the  two 
surfaces  are  opposed. 

If  the  two  infinite  parallel  planes  x  =  a,  x  =  b  carry  uniform 
currents  parallel  to  the  y  axis,  of  strength  N/kir  per  unit 
width  of  the  planes  parallel  to  the  z  axis,  and  if  the  directions 
of  the  two  currents  are  opposite,  the  region  between  the  planes 
is  a  uniform  field  of  force  of  strength  N  parallel  to  the  z  axis. 
There  is  no  force  without  the  space  included  between  the 
planes.  The  current  in  each  plane  evidently  gives  rise  to  a 
uniform  field  of  intensity  \  N  on  both  sides  of  the  plane. 

If  a  ring  surface  be  formed  by  revolving  about  the  z  axis 
an  area  in  the  xz  plane,  and  if  electricity  be  supposed  to  flow 
symmetrically  on  the  surface,  in  closed  paths  which  lie  in 
planes  through  the  z  axis,  and  coincide  with  perimeters  of 
cross-sections  of  the  ring  formed  by  such  planes,  the  field  has 
the  same  intensity  at  all  points  of  any  one  of  the  family  (/) 
of  circumferences,  the  centres  of  which  lie  in  the  z  axis  and 
have  that  line  for  their  common  axis.  If,  using  columnar 
coordinates  (?•,  0,  z),  we  denote  the  force  components  at  any 
point,  taken  in  the  directions  in  which  these  coordinates  increase 
most  rapidly,  by  H,  ®,  Z,  these  components  are  independent 
of  0.  Since  the  amount  of  work  which  would  be  done  on  a 
magnetic  pole  if  it  were  carried  around  any  closed  path  with 
out  the  surface,  —  whether  or  not  it  linked  with  the  surface,  — 
or  around  any  evanescible  path  wholly  within  the  surface, 
would  be  zero,  we  are  led  to  guess  that  the  field  outside  the 
ring  is  everywhere  zero,  and  that  the  lines  of  force  within  the 
ring  are  circumferences  of  the  /  family.  If  a  unit  magnetic 


286 


ELECTKOMAGNETISM. 


V 

FIG.  76. 


pole  were  carried  about  one  of  these  circumferences  of  radius  r, 
the  work  done  on  it  would  be  ±  2  Trr®,  and  this  is  equal  in  abso 
lute  value  to  4  nE,  where  E  is  the  whole  amount  of  electricity 
which  flows  about  the  ring  per  second.  We  learn,  therefore, 
that  ©  =  2E/r.  We  may  now  prove  that  if  there  is  no  field 
without  the  ring  surface,  and  if  the  only  component 
within  is  ®  —  2  E  /r,  the  currents  which  give  rise  to 
the  field  must  be  those  assumed  above.  The  com 
ponents  of  the  field  within  the  ring,  taken  parallel 
to  rectangular  axes,  are  —  2Ey/r'2,  2Ex/r2,  0,  so 
that  the  force  is  lamellar  within  and  without  the 
surface  of  the  ring.  To  find  what  currents  flow  in 
the  surface  itself,  we  may  use  a  circumference  s  of 
radius  r,  in  which  a  plane  perpendicular  to.  the  z  axis  inter 
sects  the  surface,  draw  two  arcs  parallel  and  very  close  to  .«?, 
one  on  either  side,  so  that  one  is  within  the  surface  and  the 
other  without  it,  and  complete  a  narrow  closed  contour  by 
drawing  two  radii  (Fig.  76)  which  make  with  each  other 
any  convenient  angle  <f>.  Only 
one  side  of  the  contour  yields  any 
contribution  to  the  line  integral 
(2  E<j>),  taken  about  it,  of  the  tan 
gential  component  of  the  force. 
This  integral  measures  the  work 
done  on  a  unit  magnetic  pole  car 
ried  around  the  contour,  and  is 
equal  to  4  TT  times  the  strength  of 
the  current  across  the  portion  of  s, 
of  length  r<f>,  which  the  contour 
encloses.  If  the  whole  flux  across 
s  is  F,  the  flux  across  this  arc  is  ^F/Zir,  and  we  have  the 
equation,  2  E<f>  =  4  7r<f>F/2  TT,  or  F  —  E, 

The  case  here  considered  is  approximately  that  of  a  ring  of 
revolution  wound  uniformly  with  fine  wire  (Fig.  77)  in  turns 
which  lie  nearly  in  radial  planes  through  the  axis  of  the  ring. 
If  there  are  n  turns  on  the  ring,  and  if  each  turn  carries  the 


FIG. 


ELECTROMAGNETTSM.  287 

steady  current  C,  E  =  n f\  and  the  force  within  the  ring  is 
2nC/r,  whatever  the  inductivity  of  the  homogeneous  soft 
medium  within  the  ring.  The  induction  in  the  medium  is 
2pnC/r,  and  the  intensity  of  its  magnetization  is  2knC/r. 
It  is  to  be  noted  that  the  reasoning  here  employed  might  be 
applied  unchanged  if  the  inductivity  of  the  medium  were  a 
function  of  r  and  z,  but  not  a  function  of  0 ;  this  would  be  the 
case,  for  instance,  if  into  air  space  within  the  ring  were  intro 
duced  a  soft  iron  ring  coaxial  with  this  space. 

A  slender  magnetic  filament  within  the  ring  surface,  of 
length  I  and  cross-section  S,  carries  2  pJES/r,  or  4  irnC /  (l/f*S) 
lines  of  induction.  The  line  integral  of  the  magnetic  force 
taken  along  a  magnetic  filament  in 
a  soft  medium  is  sometimes  called 
the  magnetomotive  force  in  it,  and 
the  ratio  of  this  quantity  to  the 
flow  of  induction  in  it  the  reluc 
tance  of  the  filament.  In  the  case 
before  us  k-n-nC  is  the  magneto 
motive  force,  and  ///i£  the  reluc 
tance.  This  last  expression  bears 
a  close  resemblance  to  the  formula 
for  the  electric  resistance  of  a  wire 
of  length  /,  cross-section  S,  and 
specific  conductivity  p..  The  reciprocal  of  the  reluctance  of  a 
magnetic  filament  in  a  soft  medium  is  sometimes  called  its 
permeance. 

If  wire  were  wound  part  way  around  a  soft  iron  ring,  in  the 
manner  described  above,  most  of  the  lines  of  induction  would 
still  be  confined  to  the  iron,  though  a  few  would  emerge  into 
the  air  at  the  ends  of  the  coil. 

If  a  radial  gap  be  cut  in  a  soft  iron  ring  completely  wound 
with  wire,  the  field  is  no  longer  symmetrical  about  the  axis  of 
2,  and  the  character  of  the  problem  is  changed.  The  line 
integral  of  the  tangential  force  taken  around  a  circumference 
inside  the  ring,  of  radius  /•  (Fig.  78),  with  centre  on  the  z  axis 


288  ELECTROMAGNETISM. 

and  plane  perpendicular  to  that  axis,  is  still  kirE  or 
but  the  portion  of  the  path  in  air  now  contributes  far  more 
than  its  due  proportion  to  the  result,  and  the  path  in  the  iron 
much  less  than  before.  We  know  that  at  any  surface  of  discon 
tinuity  in  the  inductivity  of  a  soft  medium  the  normal  com 
ponent  of  the  induction  is  continuous,  so  that  if  the  normal 
component  of  the  force  in  the  iron,  just  where  the  path  is  about 
to  emerge  into  the  gap,  is  If,  that  in  the  air  near  by  is  p.H,  where 
/u,  is  the  inductivity  of  the  iron.  Although  the  lines  of  force 
within  the  iron  are  no  longer  exactly  circular,  they  are  nearly 
so,  and  the  line  integral  of  the  force  about  the  circumference 
just  mentioned  is  very  approximately  Hr(?nr  —  <f>)  -f-  p_Hr<l>,  or 
4  TrnC ;  and  H=  2nC/\r[l  +  <£(>  -  1)  /2  TT]  J,  where  <f>  is  the 
angle  subtended  by  the  gap  at  the  axis  of  the  ring.  If,  in  the 
case  of  the  core  used,  p  =  1201,  and  if  only  one  per  cent  of 
the  ring  be  cut  away,  the  induction  in  the  iron  will  be  reduced 
to  about  one-thirteenth  of  its  old  value ;  the  reluctance  of  the 
path  will  be  increased  thirteenfold. 

If  the  lines  of  force  in  a  steady  electromagnetic  field  are  all 
circles  with  centres  on  the  z  axis  and  planes  perpendicular  to 
this  axis,  and  if  the  intensity  of  the  force  in  a  direction  linked 
right-handedly  with  the  z  axis  is/V^r2  +  y'2  or  /(?*),  it  is  evi 
dent  that  L  =  —  y  -f(r)/r,  M=x  -f(r)/r,  N=Q,so  that  u  =  0, 
v  =  0,  ±mv  =  DXM-  D,L=f'(r)+f(r)/r  =  A !>/«]/•>• 
According  to  this,  if  in  any  portion  of  the  field  f(r)  =  0,  in 
that  portion  w  is  0 ;  if  f(r)  =  c,  w  =  c/4:irr',  iff(r)  =  c/r, 
w  =  0j  and  \if(r)  =  cr,  10  =  0/2-*.  If,  on  the  other  hand, 
while  u  and  v  are  zero,  w  is  given  as  a  function  of  r,  /(?•) 

can  be  obtained  from  the  equation  /(/•)  =  —  J  rwdr;  when, 

therefore,  w  is  equal  to  the  constant  w0,f(r)=  27rw0r  +  d  /  r. 
If  the  cylindrical  surface  r  =  b  separates  two  regions  in  a  field 
of  this  kind  where  the  laws  of  force  intensity  fi(r),  /2(r) 
in  the  inner  and  outer  of  these  regions  are  different,  and  if 
/a  (&)  —  /i  (#)  =  &,  it  is  easy  to  see,  with  the  help  of  Stokes's 


ELECTROMAGXETISM. 


289 


Theorem,  that  the  surface  r  —  b  is  itself  a  current  surface  in 
which  there  is  a  total  flux  parallel  to  the  z  axis  across  any 
right  section  of  %  kb. 

Up  to  this  time  we  have  considered  only  media  which  have 
inductivities  independent  of  the  magnetizing  force.  The  per 
meabilities  of  the  so-called  magnetic  metals  do  not  in  general 
satisfy  this  condition,  and  we  ma}'  note  in  passing  that  some 
of  our  definitions  have  to  B 
be  restated  when  there  are 
masses  of  such  media  near 
a  circuit. 

If  fine  wire,  carrying  a 
steady  current,  be  wound 
uniformly  upon  a  cylin 
drical  rod  of  soft  iron, 
the  length  of  which  is  at 
least  400  times  its  diam 
eter,  the  induction  at  the 
middle  of  the  rod  is  sen 
sibly  the  same  as  if  the 
rod  were  infinitely  long, 
and  if  this  induction  be 
measured  (in  a  manner 
to  be  described  in  a  later 
section),  the  permeability  of  the  iron  may  be  determined. 
Curves  D,  E,  F  of  Fig.  79,  in  which  the  abscissas  represent 
the  magnetizing  force  H  in  units,  and  the  ordinates  the  corre 
sponding  induction  B  =  ^H  in  thousands  of  units,  show  the 
results  of  experiments  upon  specimens  of  very  soft  malleable 
iron,  soft  cast  iron,  and  very  hard  steel,  respectively.  It  is 
evident  that  so  far  from  being  constant,  the  permeability  and 
the  susceptibility  of  each  of  these  specimens  increase  to  a 
maximum  at  a  value  of  H  corresponding  to  a  point  where  a 
tangent  from  the  origin  touches  the  curve,  and  then  decrease. 
In  the  case  of  the  curve  D,  for  instance,  the  permeability,  corre- 


-E 


FIG.  79. 


290  ELECT  BOMAGNETISM. 

spending  to  a  value  OM  of  the  magnetizing  force,  is  MP  /  OM, 
that  is,  the  slope  of  the  straight  line  OP  joining  the  origin  with 
the  point  on  the  curve  which  has  OM  for  its  abscissa. 

When  the  conductors  which  make  up  a  simple  linear  circuit 
which  carries  a  steady  current  C,  and  the  soft  media  about 
it,  have  inductivities  independent  of  the  magnetizing  force, 
and  there  are  no  other  currents  and  no  permanent  magnets  in 
the  field,  the  coefficient  of  self-induction  of  the  circuit  may 
be  defined  indifferently  as  the  ratio  of  the  total  induction 
through  the  circuit  to  C  or  as  twice  the  ratio  of  the  integral  of 
/>tZr2/8  TT,  taken  over  the  field  of  the  current,  to  C2.  In  this 
case  the  magnetizing  curves  of  all  the  substances  in  the  field 
are  straight  lines,  and  these  definitions  lead  to  the  same  value 
whatever  C  is.  If  the  magnetizing  curve  of 
any  medium  in  the  field  were,  like  that  of  soft 
iron,  not  straight,  the  definitions  would  not 
agree,  and  each  would  yield  different  values 
for  different  values  of  C. 

Mechanically  soft  iron  or  steel  cannot  be 
regarded  as  magnetically  soft,  for  if  a  piece 
of  either  of  these  metals  be  magnetized  by 
FIG  80  induction,  this  magnetization  does  not  wholly 

disappear  when  the  magnetizing  force  is 
removed.  If  the  magnetizing  force  be  made  to  change  contin 
uously  from  a  given  negative  value  to  an  equal  positive  value 
and  back  several  times,  the  induction  goes  through  a  cycle 
which  may  be  represented  graphically  by  a  curve  somewhat 
like  that  shown  in  Fig.  80,  in  which  the  abscissas  represent 
magnetizing  forces,  and  the  ordinates  the  corresponding  values 
of  the  induction.  Such  diagrams  make  plain  the  fact  that 
the  induction  in  a  piece  of  soft  iron  or  steel  is  not  a  definite 
function  of  the  magnetizing  force,  and  that  the  energy  in  the 
medium,  as  defined  by  the  volume  integral  of  1/8  TT  times 
the  product  of  the  numerical  values  of  the  induction  and  the 
magnetizing  force,  may  have,  for  the  same  force,  very  different 
values,  depending  on  the  previous  history  of  the  metal.  When 


CURRENT    INDUCTION.  291 

the  induction  has  passed  through  such  a  cycle  as  that  indicated 
in  Fig.  80,  the  energy  in  the  field  returns  to  its  old  value, 
but  it  is  easy  to  prove  that  an  amount  of  work  represented 
by  1/4  TT  times  the  area  of  the  cycle  per  unit  volume  of  the 
substance  had  to  be  done  on  the  metal  during  the  cycle,  and 
that  this  appeared  as  heat.  The  reader  will  find  the  subject 
which  has  been  just  touched  upon  here  admirably  treated 
under  the  head  of  "  Hysteresis"  in  E  wing's  Magnetic  Induction 
in  Iron  and  Other  Metals,  and  in  Fleming's  The  Alternate 
Current  Transformer. 

IV.     CURRENT   INDUCTION. 

83.  Electromagnetic  Induction.  If  one  of  two  circuits 
(*lf  s2),  so  situated  that  their  coefficient  of  mutual  induction 
is  not  zero,  contains  a  galvanic  cell  and  a  key,  and  the  other 
(s2),  which  is  permanently  closed,  a  galvanoscope,  a  momen 
tary  current  appears  in  s2  when  the  key  is  depressed  so  that 
a  current  circulates  in  sl ;  and  another  momentary  current, 
opposed  in  direction  to  the  first,  runs 
through  s.2  when  the  key  is  opened  again. 
A  current  in  either  SL  or  s.2  gives  rise  to  a 
magnetic  field  and  causes  lines  of  magnetic 
induction  to  thread  s2 :  the  direction  of  the 
transient  current  in  s.2  in  each  of  the  cases 
mentioned  is  such  that  the  lines  which 
it  threads  through  s2  oppose  the  sudden 
change  in  the  flux  of  induction  through  s.2 
which  the  change  in  the  current  in  s1  tends  FIG.  81. 

to  cause.  Thus,  if  the  relative  position  of 
the  two  circuits  and  the  direction  of  the  current  in  sv  are 
correctly  indicated  in  Fig.  81,  the  transient  induced  current  in 
s2  will  flow  from  B  to  A  when  the  key  is  depressed  and  from 
A  to  B  when  the  key  is  again  opened.  In  general,  if  a  rigid, 
closed  circuit  s  is  in  a  magnetic  field  caused  either  by  perma 
nent  magnets  or  by  electric  currents  in  neighboring  circuits, 


292  CURRENT    INDUCTION. 

or  by  both  together,  and  if  the  positive  flux  Q  of  magnetic 
induction  through  any  cap  or  diaphragm  bounded  by  s  be 
changed  in  amount,  either  by  moving  s  or  by  changing  the 
field  in  any  way,  a  temporary  current  is  induced  in  s  in  a 
direction  which  tends  to  oppose  the  change  in  Q.  The  phe 
nomenon  is  quantitatively  explained,  when  s  is  unchanged  in 
form,  by  assuming  that,  superposed  upon  such  electromotive 
forces  as  the  circuit  may  already  contain,  a  temporary  electro 
motive  force  numerically  equal  to  the  time  rate  of  change  of 
Q  is  induced  in  s  in  the  proper  direction. 

Transient  currents  are  usually  induced  also  in  any  circuit  in 
a  magnetic  field  when  the  circuit  is  deformed  or  extended  in 
any  way.  These  currents,  like  those  already  considered,  are 
mathematically  accounted  for  on  the  supposition  that  there  is 
induced  in  every  circuit  element  ds,  which 
moves  in  a  magnetic  field  so  as  to  cut  across 
the  lines  of  induction  during  the  motion,  an 
'c  electromotive  force  numerically  equal  to  the 
time  rate  at  which  the  element  cuts  these  lines. 
FIG  82  This  electromotive  force  is  directed  from  the 

feet  to  the  head  of  an  observer  who,  lying 
in  the  element  and  looking  along  the  lines  of  force,  sees  these 
lines  move  past  him  from  right  to  left.  The  induced  cur 
rent  at  any  instant  in  either  direction  around  the  circuit  is 
equal  to  the  ratio  of  the  algebraic  sum  of  the  electromotive 
forces  induced  at  that  instant,  in  that  direction,  to  the  whole 
resistance  of  the  circuit.  If  in  Fig.  82  OC  represents  the 
direction  of  a  circuit  element  at  the  point  0,  OM  the  direction 
in  which  the  element  is  moved,  and  OF  the  direction  of  the 
whole  field  at  0,  the  induced  electromotive  force  will  have  the 
direction  OC,  not  CO.  The  direction  of  the  current,  induced 
by  the  motion,  in  the  direction  OM,  of  a  circuit  element  at  0 
in  a  magnetic  field  which  has  there  the  direction  OF,  may  be 
found  by  choosing  that  direction,  OC,  in  the  element  which  will 
cause  the  three  directions  OC,  OM,  OF  to  be  related  like  those 
of  the  x,  y,  z  axes  of  a  Cartesian  system.  It  is  to  be  noticed 


CURRENT    INDUCTION.  293 

that  the  direction  of  the  current  induced  in  an  element  is 
such  that  the  mechanical  action  of  the  field  upon  the  element 
carrying  this  current  alone  would  hinder  the  motion  ;  a  circuit 
element  carrying  a  current  in  the  direction  OC  in  a  field 
having  the  direction  OF  in  Fig.  83  would  be  urged  in  a 
direction  ON  perpendicular  to  the  plane  of  FOG  and  would 
move  in  the  direction  MO,  if  free  to  do  so,  rather  than  in 
the  direction  OM.  The  reader  will  do  well  to  compare,  in 
this  connection,  Figs.  68  and  82. 

If  (a,  b,  c),  (a,  (3,  y)  are  the  components  of  two  vectors,  I 
and  A,  the  vector  which  has  the  components  (eft  —  by,  ay  —  ca, 
ba  —  aft)  is  sometimes  called  their  vector  product  and  the 
quantity  —  (aa  +  bft  +  cy)  their  scalar  product.  The  vector 
product  of  I  and  A  has  a  direction  perpendicular  M 

to  the  plane  of  these  vectors  :  its  tensor  is  the 
product  of  their  tensors  and  the  sine  of  the 
angle  between  their  directions.  The  electro 
motive  force  induced  in  or  impressed  upon  an 
element  ds  of  a  linear  conductor  moving  in  a 
magnetic  field  is  evidently  equal  to  the  product 
of  ds  and  the  component  in  its  direction  of  the  vector  product 
of  the  induction  and  the  velocity  of  the  element. 

If  (Bx,  By,  7?,)  are  the  components  of  the  induction, 
(£>  *7>  0  those  of  the  velocity  of  the  element  relative  to  the 
field,  and  if  the  induction  does  not  change  with  the  time, 
the  absolute  value  of  the  electromotive  force  induced  in  the 
element  is 

l(Bs  •  -n  -  By .  0  cos  (x,  s)  +  (Bx  •  £  -  B,  •  £)  cos  (y,  s) 

+  (B, .S-Bx'ij) cos (*,  s) ] ds.     [206] 

The  whole  electromotive  force  induced  in  the  conductor  is  the 
integral  of  this  expression :  if  the  conductor  is  not  closed 
this  electromotive  force  gives  rise  to  a  statical  distribution 
of  electricity  on  the  ends  of  the  conductor,  and  hence  to  a 


294  CURRENT    INDUCTION. 

difference  of  electrostatic  potential  which  tends  to  destroy 
itself  by  causing  a  current  in  the  conductor  in  the  direction 
opposite  to  the  impressed  electromotive  force. 

If  the  induction  (Bx,  By,  Bz)  of  the  magnetic  field  in  the 
neighborhood  of  a  fixed  linear  circuit  changes  with  the  time, 
the  induced  or  impressed  electromotive  force  e  in  the  circuit 
is  equal  to  the  negative  of  the  surface  integral,  taken  over 
any  cap  S  bounded  by  the  circuit,  of 

\_DtBx  -  cos  (x,  n)  +  DtBy  •  cos  (y,  n)  +  DtBz  •  cos  («,  w)]. 
If,  then,  a  vector  can  be  found  of  which  the  vector 
(DtBx,  DtBy,  DtBz) 

is  the  curl,  then  the  line  integral,  taken  around  the  circuit,  of 
the  tangential  component  of  this  new  vector  —  increased,  if  we 
please,  by  any  lamellar  vector  —  will  be  equal  in  absolute  value 
to  the  induced  electromotive  force.  If  (Fm  Fy,  F2)  is  any  vector 
potential  (Section  69)  of  the  induction,  (DtFx,  DtFy,  DtFz)  is  a 
vector  potential  function  of  (DtBx,  DtBy,  J)tBz),  and  if  —  ^  is 
the  scalar  potential  function  of  any  lamellar  vector,  the  inte 
gral,  taken  around  the  circuit  in  positive  direction,  of 

-  \_(DtFx  +  DJ)  cos  (*,  s)  +  (DtFy  +  Drf)  cos  (y,  s) 

+  (DtFz  -f-  Djf)  cos  (»,  s)  ]     [207] 

will  be  equal  to  e.  This  value  of  the  whole  electromotive 
force  induced  in  the  circuit  will  be  obtained  if  we  assume 
that  every  circuit  element  ds  is  the  seat  of  an  electromotive 
force  equal  to  the  product  of  ds  and  the  tangential  component 
of  the  vector  -  [DtFx  +  D^,  DtFy  +  Drf,  DtFz  +  Dzfy 

If  a  closed  linear  circuit  s  in  a  magnetic  field  be  deformed 
or  moved  according  to  any  law  so  that  during  the  time  dt  the 
coordinates  (aj,  y,  z)  of  the  beginning  of  an  element  ds  receive 
increments  (&e,  8y,  8y)  which  are  analytic  functions  of  x,  y,  z, 
and  dt,  and  if,  during  the  interval  dt,  the  scalar  point  func 
tions  which  represent  the  magnitudes  of  the  components  of 
the  magnetic  induction  in  the  field  change  from  the  values 


CURRENT    INDUCTION.  295 

Bx,  By,  Bz  to  the  values  Bx',  By',  Bs',  the  flux  of  induction 
through  the  circuit  has  been  increased  by  the  amount 


C  C 


[BJ  •  cos  (x,  n)  +  £,'  •  cos(y,  n)  +  Bz'  •  cos  (2,  n] 

\_BX  •  cos  (#,  ri)+  By-  cos  (y,  n)  +  Bz  •  cos  (2,  ?i)]  dS, 

where  /S'  and  $  are  any  surfaces  which  cap  the  circuit  in  its 
final  and  initial  positions  respectively.  In  moving,  the  circuit 
traces  out  a  narrow  surface  S",  each  element  ds  of  the  circuit 
generating  the  surface  element  dS",  and  we  may  take  for  the 
cap  S'  the  surface  made  up  of  S  and  S".  We  have  therefore  d& 

=  dtC  C[DtBx  •  cos  (x,  n)  +  DtBy  •  cos (y,  ti)  +  DtBz  •  cos (2, n)~]dS 


In  the  second  integral,  cos  (2,  n)  •  dS"  measures  the  area  of  the 
projection  of  dS"  on  the  xy  plane  and  is,  therefore,  equal  to 
±  (8x  •  dij  —  By  -  dx)  plus  terms  of  higher  order  ;  the  sign  being 
positive,  if  the  direction  in  which  ds  moves,  the  positive  direc 
tion  of  the  element,  and  that  of  the  normal  to  dS"  are  arranged 
like  the  x,  y,  z  axes  of  a  Cartesian  system.  We  may  substitute 
in  the  integrand  Bx,  By,  Bz,  £  •  dt,  r,  •  dt,  £  -  dt  for  Ex',  B,,',  Bz', 
So-,  8?/,  82,  without  changing  the  value  of  the  integral,  and  then 
write 

D&  =  C\_P  •  cos  (a-,  s)+Q>  cos  (y,  s)  +  R  •  cos  (2,  5)]  ds, 

where  P  =  -  DtFx  -  D^  +  B.  •  rj  -  B,,  •  £, 

Q  =  -  DtFy  -  DJ,  +  Bx.^-  Bs-t,  [208] 


P,  Q,  R  are  said  to  be  the  components  of  the  induced  electro 
motive  force  at  the  element  ds. 

We  may  note  in  passing  that  we  cannot  generally  assume 
that  the  motion  of  the  electricity  in  a  circuit  which  is  the  seat 


296  CURRENT    INDUCTION. 

of  an  induced  current  is  governed  by  a  potential  function  due  to 
an  electrostatic  distribution  on  the  surfaces  of  the  conductors, 
or  elsewhere.  If  a  magnet,  the  axis  of  which  coincides  with 
the  axis  of  a  plane  circular  ring  of  wire,  be  made  to  approach 
or  to  recede  from  the  plane  of  the  ring,  a  transient  current 
is  induced  in  the  wire,  but  no  imaginable  electrostatic  distri 
bution  would  furnish  the  multiple-valued  potential  function 
needed  to  account  for  the  current. 

If  a  circuit  at  a  distance  from  other  circuits  and  perma 
nent  magnets  carries  a  changing  current  (7,  the  ratio  of  the 
numerical  value  of  the  intensity  of  the  electromotive  force 
induced  by  the  change  of  the  current  in  the  circuit  to  DtC 
is  sometimes  used  as  a  definition  of  the  self-inductance  of 
the  circuit.  The  mutual  inductance  of  two  circuits  may 
be  denned  in  a  similar  manner.  It  is  evident  that  all  the 
definitions  of  self  and  mutual  inductance  which  we  have 
mentioned  are  equivalent  when  all  the  media  in  the  neigh 
borhood  of  the  circuits  concerned  have  susceptibilities  inde 
pendent  of  the  intensity  of  the  field.  The  definitions  of 
this  section  are  often  used  when  there  are  masses  of  soft 
iron  or  other  magnetic  metals  near  the  circuits,  or  when  the 
circuits  themselves  are  made  of  soft  iron  conductors. 

If  a  number  of  circuits  sv  s2,  •  •  •  sn,  carrying  currents  C19 
C2,  •  •  •  Cn,  have  self-inductances  Lu  L2,---  Ln,  and  if  the  mutual 
inductance  of  sk  and  st  is  Mkl,  the  total  electrokinetic  energy 
T  is  of  the  form 


+  M12C,C2  +  M^C^  +  •  •  •  +  M2SC2C.  +  M24C2C4  -f  ... 
+  M8,CsCt  +  ...  +  MH__lt  n  Cn_,  Cn, 

where  the  L's  and  the  M'  s  are  independent  of  the  C"s  and  are 
to  be  considered  as  functions  of  a  set  of  geometrical  coordi 
nates  equal  in  number  to  the  degrees  of  freedom  of  the  system. 
If  pk  denotes  the  electrokinetic  momentum  of  sk,  that  is,  the 
partial  derivative  of  T  with  respect  to  Ck,  if  rk  represents  the 


CURRENT    INDUCTION.  297 

resistance  of  sk,  and  Ek  the  internal  electromotive  force  in  this 
circuit,  -~  measures  the  intensity  of  the  induced  electromo- 

(JLif 

tive  force,  and  T 

E"~~dt=  TkCk' 

If  the  relative  positions  of  two  rigid  circuits  slt  s2,  which 
carry  currents  C:,  C2,  and  are  surrounded  by  a  soft  medium 
in  which  there  are  no  other  conductors,  be  altered  by  changing 
under  their  mutual  action  the  geometrical  coordinate  q  by  the 
amount  dq  in  the  time  interval  dt,  leaving  the  other  coordinates 
which  determine  the  configuration  unchanged,  the  electrokinetic 
energy  T=%LV  C\2  +  MCl  C2+$L2  C}  will  receive  the  increment 

dT=  LlCl  •  dCi  +  LiCt  •  dCa  +  M(  C.2  •  d€\  +  C,  •  dC2)  +  C&  •  dM. 

The  electrodynamic  force  (in  the  Lagrangian  sense)  which 
tends  to  bring  about  this  change  of  configuration  is  the  partial 
derivative  of  T  with  respect  to  q,  taken  under  the  assumption 
that  the  other  coordinates  and  the  currents  are  constant  :  the 
work  done  during  the  change  by  this  force  is  d  W  =  C^CZ'  dM. 
Within  the  circuits  we  have 

-EiCi  •  dt  -  Cl  •d(LlCl  +  MC2)  =  C?r  •  dt, 
E^C.-dt-  Cz-d  (L2C2  +  MCI)  =  C22r.2  •  dt, 

so  that  the  work  done  against  the  inductive  electromotive 
forces  by  the  applied  electromotive  forces  (besides  the  amount 
C\r\  +  C^r2  dissipated  in  heat)  is 

+  MC2)  +  C2  •  d(L2C2 


L.C,  •  dC,  +  LZCZ-  dC2  -h  M(CZ  •  dCl  +  Q  •  dC2)  +2C1C2-  dM, 
or  dW+dT. 

If,  starting  from  rest,  the  circuits  come  again  to  rest  and  the 
currents  regain  their  steady  values  before  the  end  of  the 
interval  dt,  we  have 

^  2  dW. 


298 


CURRENT    INDUCTION. 


B/ 


The  principles  just  laid  down  enable  us  to  infer  that,  if 
the  conductor  AB  of  length  I  in  either  of  the  circuits  repre- 

B/  sented  by  Figs.  84 

and  85  be  moved 
parallel  to  itself 
along  the  rails  CB, 
DA,  in  the  direc 
tion  indicated  by 
/A  the  arrow  attached 

FIG.  84.  to  it,  with  constant 

velocity  v,  and  if 

the  field  have  the  direction  shown,  an  electromotive  force 
will  be  induced  in  AB  in  the  direction  pointed  out  by  the 
arrow  by  its  side. 
If  the  component 
of  the  total  induc 
tion  normal  to  the 
plane  of  the  circuit 
have  the  constant 
value  H  all  along  D 
AB,  and  if  r  be  the 
resistance  of  the 
whole  circuit  ABCD,  the  induced  current  will  be  IHv/r  in  abso 
lute  units.  The  volt,  ohm,  and  ampere  are  equal  respectively 
to  108,  109,  10- J  times  the  absolute  elec 
tromagnetic  c.g.s.  units  of  electromotive 
force,  resistance,  and  current  strength; 
if  in  this  example,  therefore,  I  =  1  metre, 
v  =  1  metre  per  second,  and  H  =  1,  the 
induced  electromotive  force  will  be  10,000 
units,  or  10 ~4  volts. 

If  a  Faraday's  disc  (Fig.  86)  which  has 
a  radius  a  be  rotated  in  a  uniform  field, 
in  which  the  component  of  the  induction  normal  to  the  face 
of  the  disc  is  H,  with  uniform  angular  velocity  w  in  the 
direction  indicated,  the  number  of  lines  of  induction  cut  per 


I     /A 
FIG.  85. 


FIG 


CURRENT    INDUCTION. 


299 


second  by  OP  is  %  a-Hw.  If  r  be  the  resistance  of  the  circuit, 
the  current  in  it  is  azlfat/2r  and  the  disc  is  a  very  simple 
form  of  constant  current  generator. 

Fig.  87  represents  a  circuit  a  part  of  which  consists  of  a 
rigid  wire  free  to  turn  in  the  air  about  the 
axis  of  a  magnet.  This  wire  makes  elec 
trical  contact,  by  means  of  brushes,  with 
the  magnet  at  its  mid-section  and  with  a 
conductor  which  forms  an  extension  of 
the  axis  of  the  magnet.  If  the  wire  be 
rotated  with  uniform  angular  velocity  <o, 
and  if  m  be  the  strength  of  one  pole  of  the 
magnet,  the  electromotive  force  induced 
in  the  circuit  will  be  2  m<D. 

If  a  thin  coil  (Fig.  88)  closely  embracing 
a  magnet  be  suddenly  slipped  from  one 
position  to  another,  the  electromotive  force 
induced  in  the  coil  is  proportional  to  the 
amount  of  induction  which  emerges  from 
the  surface  of  the  magnet  between  the 
two  positions. 

84.  Superficial  Induced  Currents. 
Although  a  mathematical  treatment  of  the 
currents  induced  in  a  massive  conductor  of  any  form,  in 
a  magnetic  field  varying  in  a  given  manner,  is  beyond  the 
scope  of  this  elementary  text-book,  we  may  give  a  very  simple 

proof  (taken  essentially  from  Prof.  J.  J. 

QN    ^  |  s)    Thomson's  admirable  Elements  of  Elec 

tricity  and  Magnetism)  of  the  fact  that 
the  currents  due  to  a  sudden,  finite  change 
in  the  field  lie  at  the  first  instant  wholly  on  the  surface  of 
the  conductor. 

Let  n  linear  circuits,  the  resistances  of  which  are  rlt  r2,  r3,  •  • ., 
and  the  self-inductances  Llt  L.2,  L3,  ••-,  lie  near  each  other  in 
a  magnetic  field  so  that  the  coefficient  of  mutual  induction  of 


FlG 


300  CURRENT    INDUCTION. 

the  ith  and  jth  circuits  is  Mtj.  Let  the  flux  of  the  external 
field  through  the  circuits  be  N19  N^  JV3,  •  •  -,  and  assume  that 
the  currents  are  originally  zero  and  that  no  one  of  the  cir 
cuits  contains  any  battery  or  other  generator.  If,  then,  the 
external  field  experiences  a  finite  change  during  the  extremely 
short  time  interval  r  and  thereafter  remains  constant,  the 
flux  through  the  kih  conductor  becomes  changed  from  Nk 
to  Nk.  Transient  currents,  C19  Cz,  C3,  •  •  •,  flow  through  the 
circuits  and  at  the  end  of  the  time  r  attain  the  values 
Ci,  (72',  C8'j  ••-.  During  the  given  interval  we  have  in  the 
first  conductor,  which  will  serve  as  a  general  example, 


(Mlk  -  DtCt)  +  DtN,  +  T&  =  0, 

and  if  this  be  integrated  with  respect  to  the  time  between  0 
and  T,  the  last  term  of  the  result  will  be  less  than  ^C'/r,  which 
is  negligible,  so  that  the  result  may  be  written  in  the  form 
LI  •  Ci  +  S  (Mlk  •  Ck')  +  -ZVi'  =  NI.  The  second  member  repre 
sents  the  whole  induction  flux  through  the  first  circuit  before 
the  change  and  the  first  member  the  whole  flux  at  the  end  of 
the  time  r,  so  that  the  currents  generated  by  the  sudden  change 
in  the  field  are  such  as  to  keep  unchanged  the  whole  flux. 

Imagine  a  compact  mass  of  metal  divided  into  such  cir 
cuits  as  we  have  just  considered  and  it  will  be  evident  that 
the  flux  through  every  circuit  in  the  metal  is  the  same  just 
after  the  sudden  change  in  the  field  as  it  was  before.  The 
work  done  in  carrying  a  magnetic  pole  about  any  closed  path 
in  the  metal  is  unaltered  by  the  change:  it  is  zero  before 
the  change  and  zero  after.  No  such  path  can  enclose  any 
current  filament  and,  therefore,  all  the  induced  currents  are 
initially  on  the  surface,  though  afterwards  transient  currents 
are  excited  within  the  metal.  It  is  easy  to  infer  from  this 
that,  if  the  external  magnetic  field  is  a  very  rapidly  alternat 
ing  one,  the  induced  currents  never  penetrate  very  far  into 
the  mass  of  the  conductor.  For  references  to  the  literature  of 
this  important  subject,  the  student  may  consult  Winkelmann's 


CURRENT    INDUCTION.  301 

Handbuch  der  Physik,  Vol.  Ill,  p.  403.  Various  problems  are 
discussed  at  length  in  J.  J.  Thomson's  Recent  Researches  in 
Electricity  and  Magnetism.  We  shall  confine  our  attention  in 
the  three  sections  which  follow  to  circuits  made  up  of  long 
slender  conductors  like  wires. 

85.  Variable  Currents  in  Single  Circuits.  When  a  simple 
inductive  circuit  of  resistance  r,  containing  a  constant  elec 
tromotive  force  E,  is  suddenly  closed,  the  current  in  the 
circuit  grows  gradually  in  strength  and  in  a  short  time  prac 
tically  attains  a  maximum  value  C0  =  E/r,  after  which  it 
remains  constant.  While  the  current  is  increasing  in  inten 
sity,  the  electromagnetic  energy  in  the  surrounding  medium 
—  if  there  are  no  permanent  magnets  and  no  other  currents 
in  the  neighborhood — increases  also  from 
zero  to  %LC02,  and  electrostatic  charges 
are  established  which  account  for  the 
electrostatic  potential  differences  in  the 
conductors  which  make  up  the  circuit. 
After  the  current  has  attained  the  value 
CQ  the  energy  (C^E  watts  or  C0E-  10r  ergs  per  second)  given 
up  to  the  circuit  by  the  generator  in  it  is  used  in  heating 
the  conductors  in  the  circuit,  and  EC0  =  CQ2r.  Before  the 
current  C  has  become  steady  CE  is  only  a  fraction  of  CQE, 
and  the  rate  C2r,  at  which  energy  is  used  in  heating  the  cir 
cuit,  is  a  still  smaller  fraction  of  C02?-;  hence  CE  —  (7V  is 
positive,  and  in  the  time  interval  dt  the  energy  (CE  —  C'2r)dt 
joules  is  used  partly  in  increasing  by  dw  the  energy  of  the 
electrostatic  distribution  on  the  surface  of  the  conductors  and 
elsewhere,  and  partly  in  increasing  by  d  (^  LC2)  or  LC  •  dC  the 
electrokinetic  energy  in  the  medium.  Unless  something  iu 
the  nature  of  a  condenser  is  attached  to  the  circuit,  dw  is 
usually  of  no  practical  importance,  and  we  may  write 

(CE-  C2r)dt  =  LC-dC,  or   Cr  =  E -  L  •  DtC, 
or        L.DtC+Cr  =  E,  or     C  =  E/r  +  A  -e~rt/L. 


302 


CURRENT    INDUCTION. 


It  appears  from  the  equation  Cr  =  E  —  L  •  DtC  that  the 
counter-electromotive  force  cannot  be  greater  than  E  while 
the  current  is  positive  ;  DtC>  therefore,  is  not  greater  than 
E I L  and,  unless  L  =  0,  the  current  cannot  jump  at  the 
instant  to  a  finite  value.  We  must  assume,  then,  that  C  =  0 
when  t  =  0,  so  that  C  =  E(l  -  e~t/r)  /r,  where  r  =  L/r.  The 
quantity  (1  -  e~f  /T)  has  the  values  0,  0.3935,  0.6321,  0.7769, 
0.8647,  0.9179,  0.9502,  0.9817,  0.9933,  0.9975,  0.9991  when 
the  ratio  of  t  to  T  has  the  values  0,  0.5,  1.0,  1.5,  2.0,  2.5,  3.0, 
4.0,  5.0,  6.0,  7.0.  The  difference  C  -  C0  or  -  Ee~t/T/r,  which 
we  may  call  the  induced  current,  has  the  value  —  C0  at  the 
beginning  and  sinks  to  1/eth  of  this  value  in  T  seconds,  which 

is  sometimes  called  the 
relaxation  time  of  the 
circuit.  The  induced 
electromotive  force  has 
the  value  —  J£e~t/T  and 
becomes  insignificant  in 
a  short  time  after  the 
circuit  is  closed.  The 
between  0  and  oo,  of 


FIG.  90. 


integral,   with   respect   to   the   time 
the  induced  current,  is  —  EL/  r2. 

If,  now,  the  electromotive  force  in  the  circuit  be  suddenly 
changed  to  E',  we  have  at  any  time  t  seconds  after  the  change 
E'C-dt=  C2r-dt  +  LC-dC  or  C  =  E'/  r  +  (E  —  E')e~t/T/r. 
The  induced  current  is  now  the  second  term  in  this  expression 
for  C  and  the  induced  electromotive  force  is  never  larger  than 
E-  E'.  The  quantity  e~t/T  has  the  values  1,  0.6065,  0.3679, 
0.2231,  0.1353,  0.0821,  0.0498,  0.0183,  0.0067,  0.0025,  0.0009 
when  the  ratio  of  t  to  T  has  the  values  0,  0.5,  1.0,  1.5,  2.0, 
2.5,  3.0,  4.0,  5.0,  6.0,  7.0.  It  is  to  be  noted  that  if  L  is 
expressed  in  terms  of  the  practical  unit  (the  henry)  of  self- 
induction,  which  is  equal  to  109  absolute  units,  r  must  be 
measured  in  terms  of  the  ohm,  which  is  equal  to  10°  absolute 
units  of  resistance.  The  relaxation  time  of  a  circuit,  which 
is  sometimes  called  also  its  time  constant,  is  usually  a  fraction 


CURRENT    INDUCTION.  303 

of  a  second.  The  ordinates  of  the  curve  in  Fig.  90  represent 
the  strength  of  the  current  in  the  circuit  just  described,  on 
the  assumption  that  the  electromotive  force  is  kept  constant 
for  5  T  seconds  after  the  circuit  is  closed  and  is  then  suddenly 
annihilated. 

If,  starting  with  no  current  in  the  circuit,  the  electromotive 
force  have  the  constant  value  E$  for  the  time  interval  a,  then 
the  value  zero  during  the  interval  b,  then  the  value  JE0  again 
during  an  interval  a,  then  the  value  zero  during  an  interval  I, 
and  so  on,  and  if  we  denote  e~a  /T  and  e~b  ir  by  a  and  /?,  the 
current  at  the  end  of  the  ?ith  period  of  interruption,  n  (a  H-  b) 
seconds  from  the  beginning,  will  be 

E0j3(l  -  a)  (1  +  aft  +  a2/32  +  •  •  •  +  a—1^-1)  A> 

and  the  limit  of  this,  as  n  increases,  is 


Starting  with  this  value  (70,  the  current  during  the  next 
period  a,  while  the  electromotive  force  is  equal  to  EQt  would 
be  E0(l  -  e~t/r)/r  +  C0e~t/r,  and  during  the  next  interval  b, 
when  the  electromotive  force  is  zero, 

E0(l  -  a)e-"T/r  +  C(tae~t/T. 

At  the  end  of  this  interval  the  current  is  again  <70  and  the 
state  is  final. 

If  E  in  the  equation  L  •  DtC  +  r  •  C  =  E  is  a  given  function 
of  the  time,  L-  C=e-t/T(A  +  Ce"T  -  E-dt). 

If  the  resistance  of  an  inductive  circuit  containing  a  con 
stant  electromotive  force  E  and  carrying  a  steady  current 
CQ  =  E/r  be  suddenly  changed  from  r  to  r',  we  have  at 
any  time  after  the  change  EC-dt  =  CV  •  dt  +  LC  •  dC,  or,  if 
C'  =  E/r',  C=  C'  +(C0  -  C')e~r't/L.  The  induced  electro 
motive  force  is  now  r'(C0  —  C')e-r>t/L,  and  if  r1  is  large  this 


304  CURRENT   INDUCTION. 

may  be  at  first  enormous.  Although  it  is  very  difficult  in 
practice  to  increase  the  resistance  of  a  circuit  thus  instan 
taneously,  the  rate  of  change  in  ;•  may  easily  be  made  very 
rapid,  and  the  spark  which  is  often  visible  when  a  circuit  is 
broken  bears  witness  to  the  fact  that  the  induced  electromotive 
force  is  sometimes  large. 

If  the  terminals  of  a  battery  of  internal  resistance  r  and 
electromotive  force  E  be  connected  by  a 
_|_   coil  of  resistance  rl  and  self-inductance  Llt 
9  :\    in  parallel  with  a  non-inductive  resistance 

J J     r2  (Fig.  91),  and  if  (7,  Clt  C2  represent  the 

FIG  91  strengths   of  the  currents  in  the  battery 

and  in  the  two  branches  of  the  external 
surface  respectively, 

If  the  value  of  C2  from  the  equation   before  the  last  be 
substituted  in  the  last  equation,  we  get 


r2)  =  Erz/(r  +  ra), 
where   H  =  rr^  +  rr2  +  r^,  so  that  Cv  =  Er^/  R  +  A  •  e~kt, 
where     k  =  R/Ll(r  +  ra).     C2  =  (E  -  C,r)  /  (r  +  r2). 
If  the  main  circuit  be  suddenly  closed  when  t  —  0,  we  have 


If,  after  the  circuit  has  been  closed  for  some  time  and  (7,  has 
attained  the  value  B,  the  battery  be  suddenly  detached,  Cl 
and  C2  become  suddenly  equal  numerically, 


and  d  =  Be~mi,  where  m=(rl+  r2)  /Llf 


CURRENT    INDUCTION. 


305 


If  the  poles  of  a  constant  battery  of  resistance  b  are  con 
nected  by  two  coils  in  parallel  (Fig.  92)  which  have  resist 
ances  rD  r.2  and  self-inductances  Llt  L2,  we  have 


b-C\ 


+(i  +  >x)  C,  +  b-C,  =  E, 
L.2.DtC,+(b  +  r,)  a  -  E, 


or 


b-Cl+(L.2-Dt 


r,)  C,  =  E. 


FIG.  92. 


If  we  perform  the  operations  (L2  -  Dt  -+-  b  -+-  r2)  and  6  upon 
the  two  equations  respectively,  and  subtract  one  result  from 
the  other,  we  shall  get  the  equation 


+  r2)  +  L,  (b  + 


L,  -  L,  -  D?C\  + 


whence     Cl  =  r2E/  (^  +  br.2  +  /V  2)  -h  ^  •  eA'  -h 
where  A  and  /t  are  the  roots  of  the  quadratic 
+  [A  (6  +  n)  +  £,  (6  +  7^)]  x  +  (br,  + 


=  0. 


Fig.  93  represents  a  Wheatstone's  Net  which  has  self- 
inductance  in  all  members  except  that 
which  contains  the  cell.  Using,  as 
far  as  it  goes,  the  notation  of  Section 
73,  let  us  call  the  coefficients  of  self- 
induction  of  the  branches  which  have 
the  resistances  p,  q,  ?•,  s,  g  ;  Lp,  Lq)  Lr, 
Ls,  Lg  respectively.  Let  ps  =  qr,  so 

that,  when  the  current  has  become  steady,  there  is  no  flow 

through  g,  while  the  current 

POEEE  C(q  +  8)/(p  +  q  +  r  +  s) 

flows  through  p  and  r,  and  the  current 


FIG.  93. 


306  CURRENT    INDUCTION. 

through  q  and  s.  If,  now,  the  branch  b  be  suddenly  broken, 
transient  currents  Cp,  Cffl  Cr,  Cs,  CgJ  which  have  the  initial 
values  P0,  QQJ  Pw  Qw  zero  respectively,  and  the  final  value 
zero,  will  flow  through  the  members  of  the  rest  of  the  net. 
KirchhofPs  Laws  give  at  every  instant 

p'C,  +  Lr.DtCp  +  ff.C9  +  Lg.VtCg  -q-Cq-  Lq.DtCq  =  0, 
T.  Cr  +  Lr-DtCr  -s.Cs-  Ls-DtCs  -g-Cg-  Lg-DtCg  =  0, 


If  we  multiply  each  of  these  equations  by  dt,  integrate  between 
t  =  0  and  t  =  oo,  and  write 

P  = 


=  -  Ccq-dt,  R=  Ccr-dt  = 

*/0  */0 

G=  f*Cg.dt, 

»/0 


we  shall  get  the  equations 
(p  +  q)P  + 


P  -  R  -  G  =  0. 
Whence, 


or,  since  ps  =  qr, 

c  =  Cps(Lp/p  -  Lr/r  +  L./s  -  LQ/g) 
g  (p  +  q  +  r  +  s)  +  (p  4-  q)  (r  +  s) 

If  Zg  and  is  are  both  zero  (Fig.  94),  it  is  possible  to  choose  r 
and^>  subject  to  the  condition  ps  =  qr, 
so  that  there  shall  be  no  transient 
current  through  g,  and  in  this  case 
Ltt/Lr=p/r.  If  Lqt  Lr,  Ls  are  all 
zero,  and  if  the  steady  current  C  and 
FIG.  94.  the  quantity  G  be  measured,  Lp  can 


CURRENT    INDUCTION.  307 

be  found.  This  method  of  determining  coefficients  of  self- 
induction  is  described  at  length  by  Lord  Rayleigh  in  the 
Philosophical  Transactions  for  1882. 

If  at  the  time  t  the  positive  plate  of  a  condenser  of  capacity 
A",  which  is  being  charged  by  a  battery  of  constant  electromo 
tive  force  E  (Fig.  95),  has  a  charge  Q-,  if  r  is  the  resistance  of 
the  "circuit,"  L  its  coefficient  of  self-induction,  and  C  =  DtQ, 
the  charging  current,  we  have 

E-  Q/K-  L-DtC=rC  or  L  -D?Q  +  r.  DtQ  +  Q/K=  E. 

The  general  solution  of  this  equation  for  Q  is  the  sum  of  any 

special  solution  (for  instance,  KE)  and  the  general  solution 

of  the  equation  formed  by  equating  the  first  member  to  zero. 

If.  therefore,  Xl  =  -  r/2  L  -f  R  and  X,  =  -  r/2  L  -  E,  where 

E'2  =  r2  /  4t  L2  —  1  1  'KL,  the  solution  required 

is  of  the  form  KE  -f-  ae*lt  H-  be**,  where  a  and 

b  are  constants  to  be  determined  from  the 

initial  conditions.     If  the   absolute  value  of 

the  quantity  under  the   radical   sign  in  the 

expressions    for  Xl  and  Xo  —  taken   positive, 

whatever  its  real  sign  may  be  —  is  ra2,  the  value  of  the  radi 

cal  will  be  m  or  mi  according  as  r2  is  greater  or  less  than 

4:L/K.     If  at  the  time  zero,  when   Q  =Q0)  the  circuit   be 

suddenly  closed, 


-  KE)  (X,  •  e^  -A 


,  • 


The  current  has  the  value  XjX,  (  Q0  -  KE)  (e^f  -  e&)  /  (X,  -  X^, 
and  if  Xl  and  X>  are  real,  it  has  the  same  sign  for  all  values 
of  t.  If,  however,  Xl  and  X2  are  imaginary,  the  expression 
given  above  for  Q  may  be  more  conveniently  written  in  the 
formA^-f(^0  -  KE)e~rt/-L(cos  mt  +  r/2Lm-  sin  w£),and 
the  sign  of  the  second  term  is  alternately  positive  for  ir/m 
seconds  and  negative  for  TT  /m  seconds,  so  that  the  current 
is  sometimes  positive  and  sometimes  negative.  The  curves 


308 


CURRENT    INDUCTION. 


in  Fig.  96  exhibit  Q  and  C  in  terms  of  t  in  a  case  where 
r2  >  &L/K,  QQ  =  0,  and  the  condenser  is  being  charged;  the 
curves  in  Fig.  97  correspond 
to  a  case  where  E  =  0  and 
the  condenser  is  discharging 


FIG.  96. 


FIG.  97. 


itself  through  the  circuit.     In  each  case  the  absolute  value 
of  the  current  starts  at  zero,  attains  a  maximum,  and  then 


FIG.  98. 


decreases.     Fig.  98  shows  Q  in  terms  of  t  when  E  =  0  and 
the  condenser  is   discharging  itself  through  a  circuit  such 


CURRENT    INDUCTION.  309 

that  r2  <  ±L/K)  the  curve,  the  ordinates  of  which  are  EK 
minus  the  ordinates  of  this  curve,  shows  Q  at  any  time  while 
the  condenser  is  being  charged  by  the  battery.  The  shape  of 
the  curve  may  be  seen  by  looking  at  Fig.  98  through  the 
back  of  the  leaf  and  upside  down. 

If  we  differentiate  the  equation  E  —  Q/K—L-DtC  =  rC 
with  respect  to  t,  we  get  L  •  Dt-C  +  r  •  DtC  +  C/K=  DtE  =  0, 
and  we  might  determine  C  directly  from  this  last  equation. 

If  a  condenser  of  capacity  K,  originally  charged  to  poten 
tial  QQ/KJ  be  discharged  through  a  circuit  (Fig.  99)  which 
consists  of  a  non-inductive  resistance  rl  and 
an  inductive  resistance  ?*2,  arranged  in  mul-  X 
tiple  arc,  and  if  the  currents  at  the  time  t 
through  the  branches  of  the  external  circuit  [I]  I 
be  Ci  and  C.2  respectively,  Cl  +  C2  =  -  DtQ.  I  _ 
If  we  take  into  account  the  induced  elec-  FIG.  99. 

tromotive  force,  we  may  apply  Kirchhoff's 
Laws  directly  to  this  circuit  and  learn  that  Q  /  K—  C^  =  0, 
and  that 


or  Q/K+LfDt(DtQ  +  (7,)  +  r.2(DtQ  +  C,)  =  0. 

If  the  values  C\  =  Q  /  Kr»  DtCl  =  DtQ  /  Ki\,  obtained  from 
the  first  of  these  equations,  be  substituted  in  the  last  one,  it 
becomes  L,  •  D?Q  +  DtQ(L^/Kr^  +  r,)  +  Qfa  +  r^/Kr^  =  0, 
and  the  solution  of  this  is  of  the  form  aeM  -f  be*',  where  X 
and  /A  are  the  roots  of  the  equation 

L#?  +  (Lt/Kri  +  ?-2)  x  +  (>-!  +  ra)  /  Kr,  =  0. 

After  a  and  b  have  been  determined  in  accordance  with  the 
given  conditions,  Cl  and  Cz  can  be  found  directly.  The  equa 
tion  <72r2  =  Q/K—  L«<DtCz  shows  that,  if  C2  is  positive, 
DtCi  cannot  be  greater  than  Q  /  KL»,  and  that  C2  cannot  jump 
suddenly  from  zero  to  a  finite  value  as  soon  as  the  condenser 
circuit  is  closed:  the  initial  value  of  (72  is  therefore  zero, 


310  CURRENT   INDUCTION. 

while  that  of  Ci  is  Q0/Krl}  and  under  the  conditions  of  this 
problem 

Q  =  Qo  [O^n  +  1)  eP  -  (XKr,  +  1)  e^/Kr,  (/x  -  X), 

/.  -  X), 
\n  -  X). 


If  X  and  /x  are  real,  Ci  decreases  from  the  value  Q0/  Kr^  to  zero, 
Ca  starts  at  zero,  increases  (accompanied  by  a  self-induced 
counter-electromotive  force  L2  -  DtC2,  so  that  <72r2  <  Cft)  until 
it  attains  a  maximum  at  the  time 

£  =  (logX-log/x)/(>-X), 

at  which  time  DtCz  vanishes  and  C^  =  Czr2,  and  then  con 
tinually  decreases,  accompanied  by  a  self-induced  positive 
electromotive  force,  so  that  <72r2  >  Cfo  If  we  integrate  C\ 
with  respect  to  the  time  from  t  =  0  to  2  =  oo  ,  and  remember 
that  X  +  it  =  -  (Lt  +  JTr^j,)  /^^"n,  and  that 


we  shall  obtain  the  whole  flow  QQr2  /  (r,  -f  ra)  through  rp  This 
is  the  same  (whether  or  not  X  and  p  are  real)  as  if  rz  had  no 
self  -inductance  ;  but  if  the  circuit  be  broken  before  the  dis 
charge  is  complete,  a  greater  portion  of  the  electricity  will 
have  gone  through  rx  than  would  be  the  case  if  Lz  were  zero. 
If  the  condenser  connections  have  a  considerable  resistance 
b,  the  differential  equation  becomes 

Kbr  +  br  + 


If  the  resistance  of  a  circuit  made  up  of  a  generator  of 
electromotive  force  E=f(t\  a  condenser  of  capacity  k  and 
necessary  leads,  is  r,  if  its  self-inductance  is  i,  and  if  we 


CURRENT    INDUCTION.  311 

denote  DtE  by  E\  we   have  L  •  D?C  +  r  •  DtC  +  C/K=  E'. 


If  R  =  V^/f  2  -  4  £,fir,  and 

a  =  (r/f  -R)/2  LK,  (3  =  (rK  +  7?)  /2  LK, 
the  general  solution  of  this  equation  is 

C  =  (K/K)  (eP'Cerf"  -  E'  •  dt  -  eatCe-at  •  JE'  •  dt) 


If  the  poles  of  a  battery  of  constant  electromotive  force  E 
and  internal  resistance  b  are  connected  by  a 
coil  of  resistance  i\  and  self-inductance  j 
in  parallel  with  a  condenser  of  capacity 
(Fig.  100),  we  have 

L±  •  DtCl  -f  (b  -f- 1\)  Cl  4-  b  •  C2  = 

and  E-  Q/K=bCl+(b  +  ra)Ca, 

or       6  •  D,C,  +(b  +  r9)D,C,  -f  CJK  =  0. 


If  we  perform  on  the  first  and  last  of  these  equations  the 
operations  [(b  +  r2)Dt  +  I/  A"]  and  b  respectively,  and  sub 
tract  one  result  from  the  other,  we  shall  learn  that 

r  +  br.  +  > 


and  that  C^  is  the  sum  oi  E  /(b  +  r^  and  the  general  solution 
of  the  equation  formed  by  putting  the  first  number  equal 
to  zero. 

If  the  arms  p  and  r  in  the  Wheatstone  Net  contain  con 
densers  of  capacity  Kp,  Kr  respectively,  the  steady  current 
through  q  and  s  will  be  C  =  E  /  (b  -f  q  +  s),  and  the  charges 
of  the  condensers  will  be  CqKp  and  CsKr.  If,  now,  the  bat 
tery  circuit  be  suddenly  broken,  transient  currents  will  appear 


312 


CURRENT   INDUCTION. 


in  the  remaining  members  of  the  net  and  the  condensers  will 
be  discharged.  The  whole  flow  through  p  will  be  —  CqKp, 
and  that  through  r  will  be  —  CsKr.  At  any  instant  during 
the  discharge  Cp  =  Cr  +  CgJ  and  if  we  multiply  this  equation 
by  dt  and  integrate  between  0  and  oo,  it  will  appear  that 
the  whole  flow  through  g  is  C(sKr  —  qKp).  This  will  be  zero 


86.  Alternate  Currents  in  Single  Circuits,  In  many  prac 
tical  applications  of  electricity  it  is  necessary  to  deal  with 
inductive  circuits  which  contain  periodic  electromotive  forces. 
In  the  simplest  case  the  electromotive  force  is  harmonic  of 


/ 


\ 


FIG.  101. 


the  form  Em  •  sin  (pt  —  a),  or  the  form  Em  •  cos  (pt  —  a)  ;  the 
amplitude  is  then  EQ;  the  periodj  T  =  2?r  /p\  the  frequency, 
n  =  p/2  TT  ;  and  the  phase  lag,  a. 

Two  harmonic  electromotive  forces,  of  the  same  period, 
A  -  sin  (pt  —  a),  B  •  sin  (pt  —  /3),  which  conspire  in  a  simple  cir 
cuit,  are  equivalent  to  a  single  simple  harmonic  electromotive 
force  C  •  sin  (pt  -  y),  where  C2  =  A2  +  B2  +  2  AB  •  cos  (a  -  ft) 
and  tan  y  =  (A  sin  a  +  B  sin  /?)  /(A  cos  a  +  B  cos  /?).  If  a 
parallelogram  be  constructed  with  adjacent  sides  equivalent 
on  any  scale  to  A  and  B,  and  with  the  included  angle  equal  to 
(a  —  /?),  a  diagonal  of  the  parallelogram  will  represent  C, 
and  the  angles  which  this  diagonal  makes  with  adjacent  sides 


CURRENT    INDUCTION. 


313 


of  the  parallelogram  will   be   equal  to  (a  —  y)  and  (y  —  (3) 
respectively. 

If,  starting  at  the  time  t  =  0  from  the  position  P0,  a  point  P 
be  made  to  move  uniformly  with  angular  velocity^?  in  counter 
clockwise  direction  around  the  circumference  of  a  circle  with 
centre  0  and  radius  Em,  if  Q  be  a  fixed  point  in  the  plane 
of  the  circumference  such  that  P0OQ  =  a,  and  if  y  be  any 
straight  line  in  the  plane  perpendicular  to  OQ,  the  projec 
tions  of  OP  on  OQ  and  on  y  will  be  equal,  at  any  time  £,  to 


FIG.  102. 


Em •  cos (pt  —  a)  and  Em •  sin(pt  —  a)  respectively.  If  OQ  be 
used  as  an  axis  of  real  quantities,  QOP  will  represent  the 
argument,  and  the  length  of  OP  the  modulus  of  the  complex 
quantity  Em-  e(pt~a)i ;  the  real  part  and  the  real  factor  of  the 
imaginary  part  of  this  quantity  will  be  represented  by  the 
projections  of  OP  on  OQ  and  on  y. 

If  while  P  is  moving  in  the  circumference,  y  moves  parallel 
to  itself  away  from  O  with  constant  velocity  a /Tor  ap/2ir, 
the  projection  of  P  upon  y  will  trace  out  a  sinusoid  (Fig.  101) 
the  length  of  the  base  of  which  is  a.  We  shall  frequently  find 


314  CURRENT    INDUCTION. 

it  convenient  to  imagine  diagrams  as  generated  in  this  way. 
If  in  Fig.  102  the  lines  OA,  OB,  OC  revolve  uniformly  in 
the  plane  of  the  diagram  about  0  with  angular  velocity^,  if  the 
angle  AOB  —  /?,  and  if  the  lengths  OA,  OB  are  equal  to  the 
amplitudes  of  two  simple  harmonic  quantities  a  =  am-sinpt, 
b  =  bm-  sin(^  +  )3),  the  projections  of  A,  B,  and  C  on  y  give 
the  curves  a,  b,  and  c ;  every  ordinate  of  the  sinusoid  c  is  the 
sum  of  the  corresponding  ordinates  of  the  sinusoids  a  and  b. 

If  in  Fig.  103  the  independent  lines  OA,  OB,  OC  revolve 
about  0  in  the  plane  of  the  diagram  with  the  same  constant 
r  D  angular  velocity  p,  the  lengths  of  the  pro 
jections  of  these  lines  upon  any  fixed  line 
(z)  in  the  plane  will  represent  harmonic 
quantities  of  the  same  frequency  (p/2tr), 
but  with  phase  differences  equal  to  the 
angles  between  the  lines  projected.  The 
sum  of  these  harmonic  quantities  may  be 
represented  by  the  projection  upon  z  of 
QD,  which  is  equivalent  to  the  geometric 
sum  of  OA,  OB,  OC,  if  QD  revolve  about 
Q  with  angular  velocity  p,  starting  to  move  at  the  same  time 
with  the  original  lines. 

If  a  circuit  s  which  has  a  resistance  r  and  a  self-induct 
ance  L  contains  an  electromotive  force  Em  •  cos  pt,  we  have 
L  •  DtC  +  rC  =  Em  •  cospt,  if  the  capacity  of  the  circuit  is  neg 
ligible.  The  complete  solution  of  this  equation  is  the  sum  of 
any  special  solution  and  the  complete  solution,  Ae~rt/L,  of  the 
equation  formed  by  writing  the  first  member  equal  to  zero. 
To  find  the  special  solution  needed,  we  may  consider  first  the 
equation  L  -  DtC  +  r  C  =  Em  (cos  pt  +  i  siupt)  =  Em  •  epti,  which 
is  in  some  respects  simpler;  if  any  solution  of  this  new  equation 
has  the  form  u  4-  vi,  u  is  a  solution  of  the  given  equation. 
Since  the  first  member  of  the  new  equation  is  linear  in  terms 
of '(7  and  DtC,  and  since  Dtepti  =  piepti,  it  is  clear  that  a  spe 
cial  solution  of  this  equation,  of  the  form  Bepti,  must  exist. 


CURRENT    INDUCTION. 


315 


Substituting  this  form  in  the  equation,  to  determine  £,  we 
learn  that  the  solution  is 


Em  •  e^/(r  +  Lpi),  or  Em(r  -  Lpi)e*"/(i*  + 
of  which  the  real  part  is  Em  (r  cos  pt  +  Lp  sin  pt)  /  (r3  +  L*p*),  or 

=  cos  (pt  —  a),  where  tan  a  =  Lp/r. 
The  current  in  s  is,  therefore, 


Em  •  cos(>  -  a)/  Vr2 

but  after  a  comparatively  short  time  the  first  term  becomes 
negligible,  and  then  the  current  becomes  harmonic  with  the 
same  period,  2ir/p,  and  the 
same  frequency,  p/2  TT,  as  the 
electromotive  force,  but  with  a 
retardation  in  phase  of  a.  The 
amplitude  is  ^OT/ Vr2  +  £y. 
The  radical  Vr2  +  iy  =  Z  is 
called  the  impedance  of  the  cir 
cuit  and  Lp  =  x  its  reactance, 
or  inductive  resistance,  under 
the  given  circumstances;  the 
self-induction  of  the  circuit 
reduces  the  amplitude  of  the 
current  in  the  ratio  of  r  to  Z.  The  relation  between  the  elec 
tromotive  force  and  the  current  strength  may  be  represented 
by  corresponding  ordinates  of  two  curves  like  those  shown  in 
Fig.  104. 

The  counter-electromotive  force  of  self-induction,  sometimes 
called  the  back  electromotive  force  of  self-induction,  is  equal  to 

—  L-DtC,OT    Pz   m  .  cos  (pt  -  a  -  i  TT)  ;  it  lags  90°  behind  the 

current  in  phase.  The  electromotive  force  necessary  to  over 
come  self-induction  is  the  opposite  of  this ;  it  has  the  same 
numerical  value,  but  its  phase  is  90°  in  advance  of  that  of  the 


FIG.  104. 


316 


CURRENT   INDUCTION. 


FIG.  ]05. 


current.  If  we  denote  the  amplitude,  Em/ Z,  of  the  current  by 
(7m,  the  amplitude  of  the  electromotive  force  necessary  to  over 
come  self-induction  will  be  LpCm.  Cr  is  called  the  apparent 
electromotive  force  or  the  instantaneous  energy  component  of 
the  electromotive  force;  its  ampli 
tude  is  rCm.  The  amplitude  of  the 
applied  electromotive  force  J£mcospt 
is  ZCm. 

If  a  right  triangle  be  drawn  (Fig. 
105)  the  legs  of  which  represent  r 
and  Lp  on  any  scale,  the  hypotenuse 
will  represent  Zou  the  same  scale  and 

the  angle  between  the  r  and  Z  sides  will  be  a ;  this  triangle 
is  the  triangle  of  resistances.  A  triangle  OPQ  (Fig.  106) 
similar  to  this,  the  sides  of  which  are  equal  to  rCm,  LpCm, 
and  ZCm,  may  be  called  the  triangle  of  electromotive  forces.  If 
the  figure  OQPR  be  made  to  rotate  positively  about  0  with 
constant  angular  velocity  p,  the  projections  at  any  time  of 
OQ,  OP,  OR  upon  any  line  in  the  plane  of  the  diagram  parallel 
to  the  original  position  of  OP  will  give  the  electromotive 
forces  at  that  instant. 

The  activity  o?  the  energy  spent  in  the  circuit,  during  any  time 
interval,  at  the  expense  of  the  generator  is  the  time  integral, 
taken  over  that  interval,  of  EC  =  En?  -  cos  pt  •  cos  (pt  —  a)  •  /  Z. 
The  mean  value  of  the  activity  for  any  num 
ber  of  whole  periods  is  fim2r/2(r2  -f  L*p*)t 
and  this  is  the  same  as  if  a  steady  current 
of  intensity  EJ  A/2  (r2  +  zy2)  had  passed 
through  the  circuit  during  the  interval; 
for  this  reason  Em  /  A/2  (r2  +  L2p* )  is  said  O  '<? 

to  be  the  virtual  or  effective  current.     The  FIG.  106. 

mean  values   for  any   number   of   whole 
periods  of  the  current  and  of  the  square  of  the  current  are 
zero  and   2£I*/2Z*;   the   effective   current  is,  therefore,  the 
square  root  of  the  mean  square  of  the  current,  and  this  is 
sometimes  called  the  quadratic  mean.     The  effective  applied 


CURRENT    INDUCTION.  317 

electromotive  force  is  Em  /  V2,  and  the  effective  apparent  elec 
tromotive  force  is  Emr/^/2-Z.  The  apparent  electromotive 
force  would  yield  the  current  C  if  applied  to  a  circuit  of  ohmic 
resistance  r  and  inductive  resistance  zero.  The  activity,  or 
"power  in  the  circuit,"  is  equal  for  any  number  of  whole 
periods  to  the  product  of  the  effective  current  and  the  effective 
apparent  electromotive  force.  For  this  reason  the  effective 
apparent  electromotive  force  is  frequently  called  the  effec 
tive  energy  component  of  the  electromotive  force.  The  first  term 
Em2r-cos2(pt  —  a)/Z2  of  the  second  member  of  the  equation 
EC  =  C2r  +  LC'D(C  shows  the  rate  at  which  heat  is  being 
dissipated  in  the  circuit ;  the  second  term, 

—  Em*Lp  •  sin  (pt  —  a)  •  cos  (pt  —  a)/Z2, 

the  rate  at  which  power  is  used  in  increasing  the  energy 
of  the  electromagnetic  field.  It  is  evident  that  the  average 
value  of  this  last  quantity  for  any  number  of  whole  periods 
is  zero.  The  effective  impressed  electromotive  force  is  often 
called  simply  "the  electromotive  force."  Such  voltmeters  and 
ammeters  as  are  commonly  used  in  alternating  circuits  usually 
indicate  effective  electromotive  forces  and  currents ;  their  read 
ings  must  be  multiplied  by  V2  to  obtain  the  maximum  values 
of  these  quantities. 

It  is  often  convenient,  as  Prof.  C.  A.  Adams  has  pointed 
out,  to  regard  the  values,  at  any  instant  of  the  impressed  elec 
tromotive  force  and  of  the  current,  as  the  projections,  on  the 
real  axis,  of  the  radii  vectores  which  join  the  origin  to  the  two 
points  on  the  complex  plane  which  represent  at  that  instant 
the  quantities  Em  •  epti,  Em  •  epti /  (r  +  Lpi).  This  last  expres 
sion  is  the  simple  solution  already  found  for  the  differential 
equation  L  •  Dt  C  +  r  •  C  =  Em  •  epti. 

If  in  the  problem  just  considered  we  reckon  the  time  from 
an  epoch  J  T  earlier,  we  shall  have 

E  =  Em-sm pt,     C  =  Em-sw(pt  -  a)/Z', 


318  CUKKENT    INDUCTION. 

these  quantities  may  be  regarded  as  the  projections  on  the  axis 
of  imaginaries  of  the  moduli  of  Em  -  epti  and  Em  •  epti  /  (r  +  Lpi). 
The  quantity  (r  -f-  Lpi)  has  been  called  the  complex  impedance, 
but  some  writers  give  this  name  to  r  —  Lpi. 

If  a  linear  plane  circuit  of  area  A,  resistance  r,  and  self- 
inductance  L,  in  a  uniform  magnetic  field  in  air  of  intensity  H, 
be  made  to  rotate  about  an  axis  perpendicular  to  the  lines  of 
the  field  with  angular  velocity  p,  and  if  at  the  time  t  —  0  the 
plane  of  the  circuit  is  parallel  to  the  field,  the  flux  of  the  field 
through  the  coil  at  the  time  t  is  AH  sin  pt,  and  the  current  C 
satisfies  the  equation  L-  DtC  +  Cr  —  —  pAH  cos  pt,  so  that 
after  a  few  seconds  C  =  —  HAp  •  cos  (pt  —  a)  /  Z.  The  whole 
flow  of  electricity  through  the  circuit  during  a  positive  half 
revolution  is  2HA/Z.  The  mechanical  action  between  the 
circuit  and  the  field  is  equivalent  to  a  couple  the  moment 
of  which  is  C  times  the  rate  of  change  with  respect  to  pt 
of  the  flux  AH  sin  pt  through  the  coil.  This  moment  is 
CHA  cos  pt,  or  —  H'2A^p  •  cos  pt  •  (cos  pt  —a)/Z,  its  average 
value  is  —  H2A2pr/2Z2,  and  the  work  done  against  it  in  a 
single  revolution  is  H^A^r-n-p  /  Z*.  External  work  must  be 
done  to  turn  the  coil  against  the  resistance  of  this  couple,  and 
the  equivalent  of  this  work  is  all  used  in  heating  the  circuit. 
If  the  rate  of  rotation  is  so  rapid  that  the  ratio  of  r  to  Lp 
is  small,  a  is  nearly  equal  to  -J-  rr,  and  C  is  nearly  equal  to 
—  HAsmpt/L',  CL  is  the  flux  through  the  circuit  of  the 
lines  of  its  own  field,  HA  sin  pt  is  the  corresponding  flux  of 
the  lines  of  the  external  field,  and  in  this  case  the  sum  of  the 
two  is  nearly  zero. 

If  two  points  A  and  B  in  an  inductive  circuit  be  joined  by 
an  additional  (non-inductive)  conductor  which  carries  an  elec 
tromotive  force  of  such  a  value  at  every  instant  and  so  directed 
that  no  current  ever  passes  through  the  extra  conductor,  the 
electromotive  force  may  be  taken  as  a  measure  of  the  differ 
ence  of  potential  between  A  and  B.  If  between  A  and  B  in  a 


CURRENT    INDUCTION.  319 

simple  circuit  which  carries  the  current  Cm  •  sin  (pt  —  a)  there 
is  an  ohinic  resistance  r  and  a  self-inductance  L,  the  difference 
of  potential  between  the  points  is  evidently 


where  tan  8  =  (r  •  sin  a  —  Lp  cos  a)  / (r  •  cos  a  +  Lp  •  sin  a).  If 
the  terminals  of  an  alternating  current  voltmeter  were  attached 
to  A  and  B,  the  instrument  would  measure  CmZ /  V& 

If  a  circuit  which  carries  a  current  Cm  •  eospt  contains  three 
coils  in  series  which  have  resistances  ?*15  r2,  r3  and  inductances 
LU  Lo,  Ls,  we  may  lay  off  on  a  horizontal  line  in  succession 
(Fig.  107)  the  lengths  0.4  =  r1CaaAS  =  raCm>SQ  =  rtCm.  Erect 
at  Q  a  vertical  line  and  lay  off  on  it  the  lengths 

QD  =  LlpCna  DF=LzpCna  FP  =  L3pCm. 

Then  OP  will  represent  the  amplitude  of  the  difference  of 
potential  between  0  and  P,  and  QOP  will  be  the  angle  of 
advance  of  its  phase  over  that  of 
the  current.  The  lines  «,  b,  c  rep 
resent  similarly  the  amplitudes  of 
the  differences  of  potential  of  the 
ends  of  the  separate  coils,  and  the 
angles  which  these  lines  make  with 
the  horizontal  the  phase  differences  FIG.  107. 

between  these  potential  differences 

and  the  current.  Starting  at  the  time  zero,  let  the  triangle 
OQP  revolve  about  O,  in  the  plane  of  the  diagram,  with  con 
stant  angular  velocity  p,  and  let  the  initial  position  of  OQ 
be  denoted  by  OQ0.  Let  the  points  of  intersection  of  the 
lines  a,  b  and  &,  c  be  denoted  by  G  and  H,  and  the  projections 
of  A,  B,  Q,  D,  F,  P,  G,  J7  upon  OQ0  by  corresponding  accented 
letters ;  then  the  lengths  at  any  time  of  the  lines  OG\  G'lT, 
H'P'  represent  the  instantaneous  values  of  the  "  electromotive 
forces  "  applied  to  the  several  coils,  and  the  lengths  of  OA', 
A'B',  B'Q'  the  corresponding  apparent  electromotive  forces. 


320 


CURRENT    INDUCTION. 


If   the   terminals    of  a   generator   of   electromotive  force 
Em-cospt  and  internal   resistance  r  are  connected  by  two 
conductors  in  parallel  (Fig.  108)  of  resist- 
ance  r^  r2  and  of  self-inductance  Ll}  L2 
respectively, 


FIG.  108. 


If  r  is  negligible, 


rC,  +  (r  +  r2)  C2  -  Em  .  cos  j»*. 


=  Em  •  cos  (  jp*  -  ai)  /  Vn2  4-  Ay  =  A 


cos 


and 


=       -  cos 


=      •  cos 


-  ttl), 

-  a2), 


-  a2)  /  Vr22  + 
where      tan  c^  =  Lip/rlt  tan  a2  =  L2p/r.2. 

Ct+  C2=  Cm'COS(pt-a), 

where  Cw8  =  ^2  +  B*  +  2  J7?  •  cos  (ai  -  a2) 

and       tan  a  =  (A-sin.al  +  I>  •  sin  a2)  /  (A  •  cos  at  4-  B  -  cos  a2). 

If  in  Fig.  109,  OP  =  Em  and  QOP  =  al,  OQ  =  Arv  QP  = 
A^p,  and  A  can  be  represented  by  a  length  laid  off  from 
0  on  OQ.  A  similar  construction,  represented  by  the  dotted 
lines,  may  be  made  for  B.  The  diagonal  OR  of  the  par 
allelogram,  two  sides  of  which  are 
the  lines  which  represent  A  and  B, 
represents  Cm.  If  OR  cuts  the  semi- 
circumference  in  6r,  OG  represents  the 
product  of  Cm  and  the  resistance  of 
the  divided  circuit. 


If  a  simple   harmonic  difference  of 
potential  Em  •  cos  pt  be  applied  to  two 
points  A  and  B  which  are  connected  by  n 
simple  conductors  of  resistances  rlt  r2,  r&) 
LV  L2,  L3,-",  and  impedances  Zlt  Z2,  Z3,  • 
the  n  fractions  of  the  form  rk/Zk*  be  denoted  by  F  and  that  of 
the  fractions  of  the  foimpLk/Zk2  by  G,  the  n  conductors  are 


109- 


,  self-inductances 
and  if  the  sum  of 


CURRENT    INDUCTION. 


321 


equivalent  to  a  single  conductor  of  resistance  R  =  F  /  (F~  +  G~) 
and  of  reactance  X  =  G  /  (F-  -f-  6r).  The  sum  of  the  currents 
in  all  the  conductors  is  Em  •  cos  (pt  —  a)  /  ^/  R2  +  X'2,  where 
tan  a  =  X/R. 

If  a  non-inductive  circuit  of  resistance  r  containing  a  con 
denser  of  capacity  k  and  a  generator  of  electromotive  force 
E  =  Em-sin2}t  be  suddenly  closed  at  the  time  t  =  0,  and  if  Q 
is  the  charge  on  the  positive  plate  of  the  condenser  at  the  time 
t,  E  —  Q/k  =  rC,  or,  since  C  =  DtQ, 

r-DtC  +  C/k=pEm-cospt. 
From  this  it  follows  that 

C  =  Ae~"  '*  +  Em  •  sin  (pt  +  /?)  /  V/-2  +  ra2, 
and        Q  =  B-  Arke~t/rk  +  Em  •  sin  (pt  +  /3  -  % 
where  m  =  1/pk,  and  tan  ft  = 


The  exponential  terms  soon  become  negligible,  and  if  we 
assume  that  Q  is  zero  at  the  outset,  we  shall  have  eventually 

C  =  Em  •  sin  (pt  +  ft)  /  V>~  +  m2, 


or  pEmk  -  cos(pt  —  8)  /  VI  +  k2p2r,  where  tan  8  =prk; 
Q  =  Em  -  sin 

Here  the  phase  of  the  current  is  in  advance  of  that  of  the 
applied  electromotive  force  E  by  the  angle  ft  and  in  advance 
of  Q  by  90°.  The  electromotive  forces  o 
of  the  condenser  and  generator  conspire 
in  direction  when  pt  lies  between  nv 
and  nir  -f-  a,  where  n  is  any  integer  and 
a  =  90°  —  ft ;  these  electromotive  forces 
are  opposed  when  pt  lies  between  n-n-  -f  a 
and  (n  +  1)  ir-  The  electromotive  force 
(Q/k)  necessary  to  overcome  that  of  the  condenser  lags  behind 
that  of  the  generator  by  a.  The  apparent  electromotive  force 


FIG.  110. 


322  CURRENT    INDUCTION. 

is  rC.  The  sum  of  the  squares  of  the  amplitudes  (rCm  and 
in Cm) of  r C and  Q /k  is  7£m2,  the  square  of  the  amplitude  of  E\ 
if,  therefore,  we  draw  a  right  triangle  of  which  rCm  and  w(7m 
are  legs,  the  hypotenuse  will  be  equal  to  Em,  and  the  angle 
adjacent  to  the  first  leg  will  be  /?.  If  such  a  triangle  OPQ 
(Fig.  110)  be  made  to  rotate  in  counter 
clockwise  direction,  with  constant  angu 
lar  velocity  p  about  0,  the  projections 
of  OQ,  OP,  and  OR  upon  any  line  per- 


FIG.  111.  pendicular  to  the  initial  position  of  OP 

will   give   the   apparent    electromotive 

force,  the  applied  electromotive  force,  and  the  electromotive 
force  necessary  to  overcome  that  of  the  condenser. 

If  a  condenser  of  capacity  ^  furnished  with  leads  of  resist 
ance  TI  be  joined  in  parallel  (Fig.  Ill)  with  a  condenser  of 
capacity  k2  furnished  with  leads  of  resistance  rz,  and  if  the 
compound  condenser  thus  formed  be  connected  up  with  a 
generator  of  internal  resistance  r  and  electromotive  force 
Em  -  sin  pt,  we  have 

(r  +  r$DtCi  +  r  •  DtC,  +  C,/  k,  =  pEm  .  cos  pt, 


If  r  =  0,  we  have  eventually 


cos  (pt  -  ai)  /  Vl  +  A^  V, 


<72  =  pEJt*  •  COS  (^  -  a2)  /  Vl  + 

where  tan  a!  =  pr-Jc^  tan  0%  =  przkz. 


, 


If  a  circuit  of  resistance  r  contains  (Fig.  112)    /N         lyl  L 
a  generator  of  electromotive  force  Em  •  sin  _^#,    ^r^  -  k 

a  coil  of  self-inductance  L,  and  a  condenser  of 

VTP    112 

capacity  k  in  series,  and  if  Q  is  the  charge 

on  the  positive  plate  of  the  condenser  at  the  time  t,  C  =  DtQ 

and  Em>s\-o.pt—  Q/k-  L.DtC=  Cr, 

or  L-D?C+r-DtC+  C/k       =pEm-co$pt. 


CURRENT    INDUCTION.  323 

The  real  part  of  any  solution  of  the  equation 


(and  one  evidently  exists  of  the  form  Bepti)  will  be  a  spe 
cial  solution  of  the  equation  just  formed.  It  is  easy  to  find 
B  by  substituting  Bepti  in  the  new  equation,  and  to  prove 
that  Em  •  sin  (pt  —  a)  /  R,  where  R2  =  r2  +  (pL  —  1/fcp)2  and 
tan  a  =  Qj2kL  —  1)  /pkr,  is  the  result  required.  To  obtain 
the  complete  solution  of  the  equation  for  C  we  should  need 
to  add  to  this  special  solution  the  complete  solution  (found 
in  the  last  section)  of  the  equation  formed  by  writing  the 
first  member  equal  to  zero;  this  solution  is  exponential  in 
form,  with  negative  indices  increasing  in  absolute  value  with 
the  time,  so  that  after  a  few  seconds  the  current  may  be  repre 
sented  by  the  equation  C  =  Em  •  sin  (  pt  —  a)  /  R.  It  is  to  be 
noticed  that  the  capacity  of  the  condenser  tends  to  offset  in 
some  respects  the  effect  of  the  self-induction  of  the  coil.  Since 
R2  =  r2+p2(L-l/p*k)z  and  tan  a  =p(L  -  l/p*k)/r,  it  is 
clear  that  the  current  in  the  circuit  is  the  same  as  if  the  con 
denser  were  removed  and  the  self-inductance  decreased  by 
l/p*k.  The  maximum  current  is  obtained  when  both  self- 
inductance  and  capacity  are  absent,  or  when  both  are  present 
and  such  that  Lkp2  =  1.  If  Q  =  $0when  C  has  its  maximum 
value,  the  difference  of  potential  (Q/k)  between  the  plates  of 
the  condenser  is  QQ/k  —  Em  •  cos(pt  —  a)/pRk,  and  if  the 
denominator  of  the  harmonic  term  is  less  than  unity,  this  term 
will  have  an  amplitude  greater  than  that  of  the  impressed  force. 
If  we  make  k  infinite  in  these  expressions,  they  become 
applicable  to  the  case  of  a  simple  inductive  circuit  containing 
no  condenser.  The  radical  R,  which  is  called  the  impedance  of 
the  circuit,  becomes  V?~  +  L-p2  when  k  is  infinite. 

When  an  inductive  circuit  contains  a  generator  of  electro 
motive  force  Em  •  sin  pt  and  an  electrolytic  cell  with  polariz- 
able  electrodes,  we  may  assume  that  when  the  frequency  is 


324  CURRENT    INDUCTION. 

fairly  large  the  counter-electromotive  force  in  the  cell  at  any 
instant  is  approximately  equal  to  1/Jc  times  the  quantity  of 
electricity  which  has  passed  through  the  cell  in  the  direction 
which  the  current  then  has,  since  the  last  reversal.  On  this 
hypothesis  the  cell  acts  like  a  condenser ;  k  depends  only 
upon  the  electrolyte  used  and  upon  the  size 
J  l_J  I  B  ?  and  material  of  the  electrodes.  Experiment 
'  *  shows  that  if  similar  platinum  electrodes  of 

moderate  size  be  used,  the  capacity,  per  square 


FIG  113  millimetre  of  the  surface  of  either  electrode, 

will  be  about  0.049,  0.089,  0.183,  0.049  micro 
farads,  according  as  the  electrolyte  is  a  dilute  solution  in 
water  of  K2SO^  KCl,  KBr,  or  KL 

If  between  A  and  B  in  a  simple  circuit  (Fig.  113)  which 
carries  the  current  C  =  Cm  •  sin  (pt  —  a)  there  is  a  resistance  r, 
a  self-inductance  L,  and  a  condenser  of  capacity  k  in  series  with 
the  self-inductance,  the  difference  of  potential  between  these 
two  points  is  rC  -f  L  •  DtC  +  Q/k.  If  Q  =  Q0  when  C  =  Cm, 


this  is      QQ/k  +  Cm-  Vr2  +  (Lp  -  1  /pk)*  •  sin  (pt  -  8), 
rp  •  sin  a  +  (1  /  k  —  Lp*)  cos  a 


where       tan  8  = 


rp  •  cos  a  +  (Lp*  —  1  /k)  sin  a 


If  the  ends  of  a  coil  of  resistance  i\  and  self-inductance  LI, 
which  is  joined  up  with  a  generator  of  resistance  r  and  electro 
motive  force  Em  •  sin  pt,  be  connected  by  Ej  - 
leads  of  resistance  r2  with  the  terminals  /J\ 
of  a  condenser  of  capacity  k2,  the  coil  and  ^p 
the  condenser  are  in  parallel  (Fig.  114), 
and 


Ll  •DtCl  +  (r  +  rx)  Cl  +  rC2  =  E 
td  +  (r  +  ra)DtC9  -f  C2/k2  =pEm  •  cos  pt. 


CURRENT   INDUCTION.  325 

If  r  is  negligible, 

Ci  -  EM  •  sin  (pt  -  a)/  V/V2  +  Lfp* 
and  C2  =  pEJtt  •  cos  (pt  -ft)/  Vl+AV^V, 

where        tan  a  =  L^p/r^  and  tan  /?  =  p^ky. 

In  many  practical  problems  r,  is  extremely  small,  so  that 
ft  is  negligible. 

If  the  terminals  of  a  generator  of  electromotive  force 
E  =  Em  •  sin  jit,  of  self  -inductance  L,  and 
of  resistance  r,  be  connected  (Fig.  115) 
to  the  ends  of  a  coil  of  resistance  rx  and 
self-inductance  L»  and  if  the  coil  ends 
are  attached  by  leads  of  resistance  r2  to 
the  coatings  of  a  condenser  of  capacity 
&2,  we  have  FlG- 


[(L  +  £,)  A  +  (r  +  r,)]  C,  +  (L  •  Dt  +  r)  C2  =  ^M  •  sin  ^, 
[£•.!>,*  +  r-'DJQ 

-h  [i  •  7),2  -h  (r  H-  »-2)  A  +  1  /  A-  J  C2  =  7?^  •  cos  jtf. 
If  ^  =  0,  we  have  the  case  last  considered. 

If  the  terminals  of  a  generator  of  resistance  r  and  elec- 
.  -  _  p—  -        tromotive  force  Em  •  sin  _p<  are  connected 
(\\         —~        I     I     (Fig-  H6)  by  two  conductors  in  parallel 
^rJ      k  —  L—     —  1—    having  resistances  rlt  n,  capacities  k^  k2, 
I  -      '     —  :^I        and  self-inductances  Lu  L2  respectively, 
FIG.  110.  but  no  mutual  inductance, 

L,  •  DfC,  +  (r  +  rOACi  +  r  -  AC2  +  d/^  -^m  •  cos  pt, 
L,  -  D/2C2  +  r  -  ACX  +  (r  +  >•,)  AC2  +  C2/A-2  -  ^m  -  cos      . 


If  we  apply  the  operator  \_L.2  •  D*  +  (r  +>2)A  +  1/A-J  to 
the  first  of  these  equations  and  the  operator  \_r  •  A]  to  the 


326  CURRENT    INDUCTION. 

second  and  subtract  one  result  from  the  other,  we  shall  have 
eliminated  C2  and  may  solve  for  Cl  in  the  usual  manner. 

In  a  case  which  sometimes  occurs  in  practice,  r  is  negligible 
and 

Cl  =  Hm'  sin  (pt  -  a,)/  B19       C2  =  Em.  sin  (pt  - 

where 


-#i2  =  n2  +  (PL,  -  i/klPy,  R*  =  r?  +  (PL2  - 

tan  aj  =  (p*k1L1  —  ±)/pk1r1J    tan  a2  =  (p2k2L2  —  l)/pk2r2. 

The  reader  will  find  the  subject  of  this  section  fully  dis 
cussed  in  Bedell  and  Crehore's  Alternating  Currents,  Franklin 
and  Williamson's  Elements  of  Alternating  Currents,  Steinmetz's 
Alternating  Current  Phenomena,  Heaviside's  Electrical  Papers, 
and  in  many  other  books. 

87.  Variable  and  Alternate  Currents  in  Neighboring  Cir 
cuits.  If  the  coefficients  of  self-induction  of  two  neighboring 
circuits  slt  s2>  —  which  contain  constant  generators  the  elec 
tromotive  forces  of  which  are  EI  and  E2  respectively,  —  are 
LI,  L2,  and  their  coefficient  of  mutual  induction  M,  if  the 
resistances  of  the  circuits  are  rlt  r2,  and  the  currents  which 
pass  through  them  at  the  time  t  are  Cl}  C2,  then 

Ll  •  Dtd  +  M.  DtC2  +  T&  =  EI, 
M-  DtCl  +  L2  •  DtC2  +  r2C2  =  E2. 
It  is  to  be  noticed  that,  since  the  electrokinetic  energy 


or  i 

must  always  be  positive  whatever  the  values  or  directions  of 
the  currents,  if  they  exist  at  all,  L^2  —  M2  can  never  be  nega 
tive.  If  we  substitute  in  the  differential  equations  just  found 
C7/  and  C2  for  Cl  and  C2,  where 

C,'  =  C,  -  E./r,,    C2'  =  C2  -  JE2/r2, 
we  get 


CURRENT   INDUCTION.  327 

or,  symbolically  written, 


If  we  perform  the  operation  (L2  •  D  +  r2)  on  the  first  of  these 
equations  and  the  operation  (M-Dt)  on  the  second  and  sub 
tract  one  of  the  resulting  equations  from  the  other,  we  shall 
eliminate  C.2'  and  get  the  homogeneous  linear  equation 

(L,L,  -  J/2)  DfCJ  +(r*Li  +  r^D^  +  r^C^  0. 

The  general  solution  of  this  equation  is  of  the  form  A^  +  B^e"', 
where  X  and  n  are  the  two  roots  of  the  equation 


that  is, 


Lo)  ±      (  IV&!  +  jyLa)  -  -  4 


2  (L,L.2  -  M-) 

If  we  eliminate  C/  from  the  original  equations,  we  shall  learn 
that  CV  =  ^2eA'  +  B^  where  X  and  p  have  the  values  just 
given.  "  Both  \  and  /x  are  negative,  since  Z^  -  M  is  positive, 
and  both  are  real,  since  the  expression  under  the  radical  sign 
may  be  written  (L&  -  L^)  -  +  Ir^M  ~.  The  coefficients 
Av  Av  BV  B,  in  the  expression  for  CV,  C2'  are  not  all  inde 
pendent,  for  we  find  when  we  substitute  these  expressions  in 
either  of  the  original  equations  that  the  ratios  A2/Al}  B2/  B± 
must  have  the  fixed  values 


r2)    or  -  (L,\  + 
and  -  MI*./  (L^n  +  r,)  or  -  (i^  +  rJ/Mp. 

respectively.  If  we  denote  these  ratios  by  a  and  /3,  we  have 
C,  =  E,  /  r/H-  A^  +  B^,  C2  =  E«/r2  +  a  A^  +  QB^',  where 
X,  /A,  a,  ^8  depend  only  upon  the  forms  of  the  circuits  and 
the  materials  of  which  they  are  made  and  A^  Bl  are  to  be 


328 


CURRENT    INDUCTION. 


determined  from   the  conditions  of  the  particular  problem 
under  consideration. 

If  E2  =  0  (Fig.  117),  and  if  sl  be  suddenly  closed  at  the 
time  t  =  0,  Cv  and    C2  are  initially  zero, 


FIG.  117. 


and 


Al-\-Bl  =  -El/rly 

'^{1-i^'COv^- 


lltt  4-^/3  =  0, 


1] 


where  R  stands  for  the  radical  in  the  expressions  for  X  and  /A. 
The  time  integral  of  <72  from  0  to  oo  is  evidently 


and  is  the  same  whichever  circuit  contains  the  given  electro 
motive  force  and  is  used  as  the  primary.  Fig.  118  shows  the 
currents  in  sl  and  s2  under  these 
circumstances,  when 

-JTJ     f\        T       1         T       __     1 


If  E%  =  0,  and  if  slt  which  has 
been  closed  for  some  time,  has 
its  resistance  suddenly  changed, 
when  t  —  0,  from  r0  to  r15  we  have 
initially  Cl  =  El/rQ  and  C2  =  0, 
so  that 


A,  =  I3E,  (rQ  - 
and  C2  = 


^  (a  -  0), 
x  -  r0 


CURRENT    INDUCTION. 


329 


The  integral  of  C2  with  respect  to  the  time,  from  0  to  <x>,  is 
E^Ifa  —  fo)/rorir2>  and  the  limit  of  this,  as  -rL  grows  larger 
without  limit,  is  E^I/rfa.  C2  attains  its  maximum  value  at 
the  time  (log  X//A)/(/A  —  X)  :  this  fraction  approaches  zero 
when  rL  increases  without  limit.  If  t\  is  infinite,  we  have 
Cl  =  0,  6 2  =  Ae~r*/L2:  the  time  integral  of  C2,  between  0 
and  oo,  is  AL2/r2,  from  which  we 
infer  that  A  =  E^M/r^L^. 

If,  then,  i\  is  infinite,  that  is,  if 
the  circuit  is  suddenly  broken,  the 
current  in  the  secondary  jumps 
instantly  from  zero  to  the  value 
E^I/r^L.2  and  then  decreases 
after  the  manner  shown  in  Fig. 
119,  which  is  drawn  to  scale  on 
the  assumption  that  in  practical 
units  El  =  2)  L2  =  %,  J/=  I/  V8,  r0=l.  The  whole  area 
between  the  time  axis  and  the  curve  which  represents  the 
current  in  s2  is  the  same  in  Fig.  119  as  in  Fig.  118,  though 
the  shapes  of  the  curves  are  very  different. 

If  E2  =  0,  and  if  E1  =  Em  -  cos  pt,  we  have 


FIG.  119. 


and 


M-  DtC2 
L2  •  DtC2 


?-1C1  =  Em  •  cospt 
rC  =  0. 


If  Cl  =  x,  C2  =  y  represent  any  special  solution  of  these 
equations,  the  complete  solution  may  be  found  by  adding 
x  and  y  to  the  values  of  Cl  and  (72,  found  earlier  in  this  sec 
tion,  which  completely  solve  the  equations  formed  by  equating 
to  zero  the  first  members.  These  last  quantities,  however,  are 
exponential  in  form  with  indices  intrinsically  negative ;  after 
a  few  seconds  they  are  negligible,  and  we  need  use  only  the 
final  forms  of  x  and  y.  The  real  parts  (?/1?  i/2)  of  any  solution, 
of  the  form  C^  =  MX  -f  v^,  C2  =  n.2  +  ?y,  of  the  equations 


330  CURRENT    INDUCTION. 

form  a  solution  of  the  original  equations.  Applying  the  opera 
tor  (L2  •  Dt  -f-  r2)  to  the  first  of  these  new  equations  and  the 
operator  (  M  •  Dt)  to  the  second  and  subtracting  one  result  from 
the  other,  we  get 


(L,L2  -  M^DfC,  +  (r,L2  +  r.L^D.C,  + 


an  equation  which  evidently  has  a  solution  of  the  form  B  •  epti, 
and  if  we  substitute  this  expression  in  the  equation,  we  learn 
that 


B  =  %m  (r2  +  L2pi)  I  [n 
The  real  part  (x)  of  Bepti  is,  therefore, 


Em  •      £2y  +  r2*  •  cos  (pt  +  8  -  ff) 


where       tan  8  =  L2p/r2  and 

tan  0  =  (r-gLi  +  r^)  p  /  [r^  —  (LrL2  — 
or  x  EEE  ^4  •  cos  (_p^  —  a),  where,  if 

L  =  L,-  M  2L2lr  I  (L./p2  +  ?'22) 
and  r  =  n  +  M*p>rs/(L2*p*  +  n2), 


^4  =  Em/  ~\  L2p-  +  v-2,  and  tan  a  =  Lp  /r. 

The  primary,  therefore,  behaves  like  a  single  circuit  (at  a 
distance  from  all  others)  of  resistance  r,  greater  than  r1?  and 
of  self-inductance  L,  less  than  Lr  The  presence  of  the 
secondary  circuit  makes  the  lag  in  the  primary  less  than  it 
would  otherwise  be. 

The  corresponding  value  (?/)  of  C2  can  be  found  by  substi 
tuting  x  in  one  of  the  original  equations  :  it  is 


•  cos  (pt  -ft)/  -^L/p2  +  r22, 
or  MpA  .  cos  (pt  +  TT  -  ft)  /  VZ2y  +  r22, 


CURRENT   INDUCTION. 


331 


where  tan  (fi  —  a)  =  r.2/L2p.    The  lag  in  phase  of  the  secondary 

circuit  behind  the  primary  is  TT  +  a  —  /?,  or  \  IT  +  tan"  l  (L2p/r2). 

The  lag  of  the  secondary  current  behind  the  electromotive 

force  is  tan-  1  [^  (L,L,  -  M*)  -  >y-2]  /|>  (r2L,  +  r^)].     The 

average  rate  for  any  number  of  whole  periods  at  which  the 

generator  furnishes   energy  to  the 

primary   is    the    average    value    of 

EmA  •  cos  pt  •  cos  (pt  —  a),    which   is 

i  EmA  •  cos  a  or  Em*r  /2  (L*p*  +  r2)  ; 

this  is  greater  when  the  secondary 

is  closed  than  when  it  is  open.     The 

average  rate  for  any  whole  number 

of  periods  at  which  energy  is  used  in 

heating  the  secondary  is  the  average 

value  of  C22  ?%  or  r2M'2p*A2/2(L22p*  +  r22)  ;  the  ratio  of  this  to 

the  power  used  in  the  primary  is  called  the  efficiency  of  the 

transformation  and  is  equal  to  rJlty/r(Lf]P  +  r£).      The 

electromotive  force  induced  in  the  secondary  is 


FIG.  120. 


The  problem  here  considered  is  in  principle  that  of  the  alter 
nate  current  transformer  (Figs.  120  and  121),  and  it  is  fre 
quently  the  case  in  practice  that  the  ratio 
of  /•;,  to  L2p  is  very  small.  Under  these 
circumstances  L,  r,  the  amplitude  of  C2, 
and  (3  —  a  are  nearly  equal  to 

LI  -  M*/L2,  r,  +  rJIP/io2,  MA/L&  and  0 

respectively.  Both  circuits  are  usually 
wound  on  a  soft  iron  core  (often  a  ring) 
of  great  permeability,  and  L-^LZ  —  M*  is 
very  small  compared  with  either  L^  or  L.,  ; 
in  this  case  the  lag  of  the  primary  is  negligible,  while,  for 
high  frequencies,  that  of  the  secondary  is  nearly  two  right 
angles.  If  L^L.2  —  J/2  is  practically  nothing,  the  transformer 
is  said  to  have  no  magnetic  leakage.  The  ratio  of  L^  to  L2 


FIG.  121. 


332  CURRENT    INDUCTION. 

is  usually  nearly  equal  to  that  of  the  square  of  the  number 
of  turns  (n^/n^)  of  the  circuits  on  the  core,  and  under  these 
circumstances  B  is  approximately  equal  to  n^A/n^.  For 
exhaustive  treatments  of  the  problem  of  this  section,  which 
is  of  much  practical  importance,  the  reader  is  referred  to  such 
books  as  Fleming's  The  Alternate  Current  Transformer;  J.  J. 
Thomson's  Elements  of  Electricity  and  Magnetism  ;  Nipher's 
Treatise  on  Electricity  and  Magnetism;  and  Steinmetz's  Alter 
nating  Current  Phenomena. 

88.   The  General  Equations  of  the  Electromagnetic  Field. 

When  a  fixed,  metallic,  linear  circuit  s  of  specific  conductivity 
X  =  l/o-,  at  a  uniform  temperature  throughout,  carries  an 
induced  current,  positive  electricity  is  urged  around  s  in  the 
direction  of  the  current  by  "  something  of  the  nature  of  an 
electrostatic  field,"  though  we  do  not  need  to  assume  that  this 
is  always  due  to  electrostatic  charges.  If  we  denote  the  com 
ponents  of  the  field,  at  every  point  within  or  without  the 
conductors  which  form  the  circuit  by  X,  Y,  Z,  the  line  integral 
of  [A'  •  cos  (x,  s)  +  Y-  cos  (y,  s)  +  Z-cos  (z,  «)],  taken  around  s 
in  the  direction  of  the  current,  is  the  internal  electromotive 
force  and  is  equal  to  the  negative  of  the  time  rate  of  change 
of  the  positive  flux  of  magnetic  induction  through  the  circuit. 
If  the  circuit  be  covered  by  a  cap  S,  if  n  denotes  the  direction 
of  the  normal  to  S  drawn  towards  the  positive  side,  and  if 
Bx,  By,  Bz  are  the  components  of  the  magnetic  induction  B, 
then,  on  the  assumption  that  Stokes's  Theorem  may  be  applied 
to  the  vector  (X,  Y,  Z),  we  shall  have 


=  ~  JJ 


Y)  cos(x,  n)  +  (DZX  -  Dx^)cos  (y,  n) 

+  (DXY-  DyX)  cos  (»,  7i)  ]  dS 
cos  (x,  n)  -f  DtBy  •  cos  (y,  n) 

+  DtBz-COS(z,ri)']dS, 


so  that  the  expression 

-  Dz  Y+  DtBx)  cos  (x,  n)  +  (DZX  -  DXZ+  DtBy)  cos(y,  n) 
+  (DXY-  DyX+  DtBz)  cos  (*,  n)] 


CURRENT    INDUCTION.  333 

integrated  over  any  cap  bounded  by  s,  whatever  the  forms  of 
the  latter,  yields  zero.  We  are  led  to  assume,  therefore,  that 
at  every  point  within  or  without  any  such  circuit 


=  DXY-DVX,  [209] 

and  to  say  that  the  negative  of  the  vector  the  components  of 
which  are  the  time  derivatives  of  the  component  of  the  induc 
tion  is  equal  to  the  curl  of  the  electric  field. 

If  £,  77,  £  are  the  components  of  the  curl  of  the  magnetic 
induction  B,  and  if  the  components  Fx,  Fy,  Fz  of  the  vector 
F  are  defined  by  the  equations  4  irFx  =  Pot  £,  4  irFy  =  Pot  77, 
4  irFz  =  Pot  £,  F  is  a  vector  potential  function  of  B.  By  its 
aid  we  can  transform  the  integral 

-  CC[DtBx  •  cos  (x,  ?i)  +  DtBy  •  cos  (y,  n)  +  DtBz  •  cos  (z,  n)~\  dS, 

in  which  the  integrand  is  the  component  normal  to  S  of  the 
curl  of  DtF,  into  a  line  integral  taken  about  5  of  the  tangen 
tial  component  of  DtF.  We  have,  therefore, 

\  \_X-  cos  (x,  s)  +  Y-  cos  (y,  s)  +  Z •  cos  (z,  s)~\ds 
=  -  J"[  A^«  •  cos  (x,  s)  4-  DtFy  •  cos  (y,  «)  +  DtFz  •  cos  (z,  s)]  ds, 

and  the  integrands  can  differ  only  by  the  tangential  com 
ponent  of  some  lamellar  vector  (Gx,  Gy,  Gz),  which  adds  nothing 
to  the  integral  taken  completely  around  s.  Since  this  is  true 
whatever  the  shape  of  s,  we  assume  that  at  every  point 

X=  —  DtFx  -h  Gx,    Y  =  —  DtFv  +  Gy,    Z  =  —  DtF.  +  Gz. 

When  the  magnetic  field  is  constant  and  the  components  of 
DtF  vanish,  X,  Y,  Z  are  equal  to  -  Dx  F,  -  Dy  F,  -  Dt  F,  and 


334  CURRENT    INDUCTION. 

the  phenomena  will  be  accounted  for  if  we  follow  Maxwell 
and  write 


Z=-DtFz-DeV.  [210] 

The  reader  should  compare  these  equations  with  [208]. 

Within  the  conductors  which  form  s,  the  components 
(u,  v,  w)  of  the  conduction  current  (q)  satisfy  Maxwell's  cur 
rent  equations 

4  TTU  =  DVN  -  DZM,   4  TTV  =  Dz  L  -  DXN, 

±Ttw  =  DxM-  DyL,  [211] 

where  L,  M,  N  are  the  components  of  the  magnetic  field,  and 
u  =  XX,  X  =  cru,   Y=<rv,   2=  vw. 

According  to  Poisson's  hypothesis,  a  dielectric  consists  of 
perfectly  conducting  molecules  separated  from  each  other  by 
perfectly  insulating  spaces,  the  specific  inductive  capacity  (K) 
depending  merely  upon  the  ratio  of  the  volumes  of  the  spaces 
occupied  by  the  molecules  and  the  intervening  spaces.  From 
this  point  of  view,  there  is  a  transfer  of  electricity  through 
every  molecule  when  the  dielectric  is  being  polarized,  one 
portion  of  the  surface  of  the  molecule  becoming  positively 
electrified  by  induction  and  another  portion  negatively  elec 
trified.  Every  change  in  the  polarization  is  accompanied  by 
the  passage  of  electricity  through  the  mass  of  the  molecule, 
and  we  are  to  assume  that  during  the  change  every  molecule 
acts  electromagnetically  like  a  current  element.  Whatever 
our  theory,  the  appearance  of  the  induced  charges  which 
account  mathematically  for  the  phenomena  observed  when  a 
dielectric  becomes  polarized,  involves  the  displacement  of 
electricity,  and  corresponding  electromagnetic  effects.  In  his 
famous  paper  on  "A  Dynamical  Theory  of  the  Electromag 
netic  Field,"  published  in  the  Philosophical  Transactions  of 
the  Royal  Society  in  1864,  Maxwell  assumed  that  whenever  the 
polarization  of  a  soft  dielectric  in  which  the  electric  induction 
has  the  components  ®x,  <£,,,  &z  is  being  changed,  electromagnetic 


CURRENT   INDUCTION.  335 

phenomena  are  to  be  looked  for  equivalent  to  those  which 
would  accompany  the  presence  of  currents,  called  displacement 
currents,  in  the  dielectric  denned  at  each  point  by  the  vector 


or  (K-DtX/litj    K.DtY/±ir,    K- 

According  to  this  assumption, 

u1  =  -#,<*>*  /4  T  H-  \X,      v'  =  D&y  /4  TT  +  A  F, 
w'  =  Dt<S>x/±ir  +  \Z, 

where  u',  v',  w'  are  the  components  of  the  total  current,  and 
we  may  write  the  current  equations  in  the  generalized  form 

4  TTU'  =  D&x  +  4  ,rw  =  DyX  -  DM, 
4  TTV'  =  D&y  +  4  TTV  =  DZL  -  DXN, 
4  TTW'  =  D&z  +  4  TTIV  =  DJl  -  DyL,  [212] 

in  which  u,  r,  w  represent  the  components  of  the  conduction 
current  alone.  In  conductors  the  displacement  currents  are 
negligible,  in  a  perfectly  insulating  dielectric  the  conduction 
currents  vanish  ;  both  are  supposed  to  coexist  in  dielectrics 
which  are  slightly  conducting.  Within  a  conductor,  since  the 
curl  of  the  magnetic  force  is  solenoidal,  Dxu  +  Dyv  +  Dzw  =  0. 
If  at  least  that  portion  of  the  magnetic  induction  near  the 
current  which  changes  with  the  time,  is  induced  in  soft  media, 
and  if  /x  is  the  magnetic  inductivity  at  the  point  (a?,  y,  z), 
we  have  DtBx  =  ^  •  DtL,  DtBy  =  /i  •  DtM,  DtBz  =  ^  •  DtN,  and 
[209]  becomes 

-^•DtL  =  DyZ  -  DZY,    -  /x  •  DtM=  DZX-  DXZ, 

-^.DtN=DxY-DyX,  [213] 

or,  if  the  media  are  homogeneous, 

-  fjt\  •  DtL  =  Dyw  -  Dzv,    -  ftX  •  DtM  =  Dzu  -  Dxw, 

-  p.\  -  DtN  =  Dxv  -  Dyu.  [214] 


336  CURRENT    INDUCTION. 

If  we  differentiate  the  equations  of  [212]  with  respect  to 
t  and  substitute  the  values  of  DtL,  DtM,  DtN  from  [214] 
in  the  results,  we  shall  get  for  homogeneous  media  three 
equations  of  the  form 


DfX  +  4  TT  •  Dtu)  =  Vzu  -  Dx(Dxu  +  Dyv  +  Dzw)  =  V2w, 
that  is, 

/*\(JST.  7>taJT  4-  4  u  -  Z><tt)  =  VX    M(^-  A2  y  +  4  TT  -  Z><V)  =  v«w, 
A*X(JT-  Z>,2^  +  4  TT  -  D,M;)  =  V2w.  [215] 

Where  there  is  no  conduction  current  these  become 
pK'D*X=V*X,  nK.D?Y=^Y,  pK-D?Z=V*Z.  [216] 

If  we  substitute  in  the  equations  [214]  the  values  of  u,  v, 
and  w  from  [211],  we  shall  obtain  for  homogeneous  media 
the  equations 


4  TT/A  •  DtL  =  V2Z,       4  7r/xA  -  DtM  =  V2M, 

4  TT^tA  •  DtN  =  V2N. 
The  energy  of  the  field  is  W+  T  where 


MISCELLANEOUS   PROBLEMS. 

1.  The    astronomical   unit    of   mass    in  any   length-mass- 
time  system  is  the  mass  which,  concentrated  at  a  fixed  point, 
would  cause  by  its  attraction  unit  acceleration  in  any  particle 
at  the  unit  distance.     The  astronomical  unit  of  mass  concen 
trated  at  a  point  at  a  unit  distance  from  a  particle  of  mass 
equal  to  the  absolute  unit  would  attract  it  with  a  force  of  one 
unit.     Show  that  the  astronomical  unit  of  mass  in  the  c.g.s. 
system  is  15,430,000  grammes,  while  in  the  f.p.s.  system  it  is 
963,000,000  pounds.    Show  also  that  the  mass  which,  concen 
trated   at  a  point  distant  1   centimetre  from  a  particle  of 
equal  mass,  would  attract  it  with  a  force  of  1  dyne,  is  only 
3928  grammes.     Prove  that  the  earth's  mass  (Problem  9)  in 
astronomical  c.g.s.  units  is  3.98  x  1020.     Show  that  a  mass  of 
1  kilogramme  must  be  raised  about  3  metres  at  the  earth's 
surface  in  order  to  reduce  its  weight  by  1  dyne. 

2.  Prove  that  two  equal  marbles,  each  of  4  grammes  mass, 
must  be  placed  with  centres  a  little  over  1  centimetre  apart,  if 
the  attraction  between  them  is  to  be  1  microdyne,  and  find  the 
attraction  [5535  kir~]  of  an  iron  cylinder  of  revolution,  of  10  cen 
timetres  radius,  1  metre  long,  upon  a  marble  of  100  grammes 
mass,  with  centre  in  the  axis  of  the  cylinder  and  distant  10 
centimetres  from  the  nearer  base.     If  the  specific  gravity  of 
iron  is  7.5,  the  radius  of  each  of  two  equal  iron  balls  which, 
placed  in   contact,  attract  each  other  with  a  force    of   one 
gramme's  weight  is  88.5  centimetres.     If  the  mass  of  each  of 
two  equal  homogeneous  spheres  with  centres  1  mile  apart 
were  415,000  gross  tons,  the  attraction  between  them  would 

337 


338  MISCELLANEOUS    PROBLEMS. 

be  about  1  pound's  weight.  The  force  of  attraction  between 
two  equal  particles  1  foot  apart  and  each  of  mass  n  times  as 
great  as  that  of  a  cubic  foot  of  water,  would  be  equal  to  the 
weight  of  about  ?i2/(7.94  x  IQ6)  pounds. 

3.  Assuming  that  a  force  equivalent  to  the  weight  of  a 
mass  of  1  gramme  is  equal  to  427r2(98.95)4  centimetre-gramme 
attraction  units,  find  the  radii   of  two  equal  homogeneous 
spheres  which,  made  of  matter  of  density  6,  would  attract  each 
other  with  a  force  of  1  gramme's  weight  if  they  were  placed 
in  contact  with  each  other.     [98.95.] 

4.  Assuming  that  1  dyne  is  equal  to  15,430,000  absolute 
c.g.s.  attraction  units  and  that  1  poundal  is  equal  to  13,825 
dynes,  show  that  if  two  equal  homogeneous  spheres  of  density 
p,  when  placed  in  contact,  attract  each  other  with  a  force  of 


/dynes,  the  radius  of  each  is  about  (43.3)  'y     cm.,  and  that 

two  equal  homogeneous  spheres  of  the  density  of  water  when 
in  contact  will  attract  each  other  with  a  force  of  1  dyne,  1 
gramme's  weight,  1  poundal,  or  1  pound's  weight,  according 
as  the  radius  of  each  in  centimetres  is  43.3,  242.2,  469.4, 
or  1118.5. 

5.  Show  that,  having  found  the  value  of   the  attraction 
unit  of  force  in  any  length-mass-time  system  in  terms  of  the 
absolute  unit  of  force  in  this  system,  you  may  find  the  value 
of  the  attraction  unit  of  force  in  any  other  system  the  ratios 
of  the  fundamental  units  of  which  to  those  of  the  old  system 

2 

are  A,  /x,  and  r,  by  multiplying  the  found  value  by  *~ 

A 

6.  Show  that  if  two  homogeneous  spheres  of  mass  ml  and 
ma,  starting  from  rest  with  centres  at  a  distance  a  apart,  move 
toward  each  other  under  their  mutual  attraction,  and  if  at  any 
time  t,  x  represents  the  distance  between  the  centres, 


A2*  =  — v  \        '  D*  -  -:. , 

X  {*•«*' 


MISCELLANEOUS    PROBLEMS.  339 

=  \.->  7  /    ('  .  -  7  1  Va^  (a  -  x)  +  tf  cos-'V-  }• 
*2  k  (M!  +  ?>?2)    [  *aj 


=  \L  7  ,    a  ,  -  r  I  Vz  (a  -  x)  +  a  tan^xp^  i 
^  2  &  (»&!  +  w2)    L  *     a:      J 


Hence  prove  that  if  the  spheres  are  each  one  foot  in  diam 
eter  and  of  density  equal  to  the  earth's  mean  density,  and  if 
their  surfaces  are  i  of  an  inch  apart  at  the  start,  they  will 
come  together  in  about  five  minutes  and  a  half.  In  this  con 
nection  we  may  note  that  if  M  is  the  mass  of  the  earth,  R  its 
radius,  p  its  mean  density,  and  k  the  gravitation  constant  for 
the  particular  units  used, 

kM  3g 

"-^^^  =  4^k' 

If  the  first  sphere  is  fixed  while  the  second,  of  mass  m2,  is 
free  to  move, 


ax 

\x 
a 


If  in  this  case  the  radius  of  the  fixed  sphere  is  r,  and  if  mz 
is  comparatively  small  and  a  infinite,  the  velocity  with  which 
the  second  sphere  reaches  the  surface  of  the  first  is  some 
times  called  the  final  velocity  for  bodies  falling  to  the  fixed 


sphere.     Its  value  is  \- -,  or  V2/-  /•,  where  f  is  the  force 

of  gravitation  at  the  surface  of  the  fixed  sphere. 

Show  that  if  the  diameter  of  the  sun  is  109.4  times  that 
of  the  earth  and  its  mass  331,100  times  the  earth's  mass, 
the  final  velocity  for  bodies  falling  into  the  sun  is  55  times  the 
final  velocity  for  bodies  falling  into  the  earth.  The  radius 
of  the  earth  being  6.37  X  108  centimetres,  show  that  the 
final  velocity  for  bodies  falling  to  the  earth  under  the  attraction 


340  MISCELLANEOUS    PROBLEMS. 

of  the  earth,  only  is  nearly  11,180  metres  (or  about  7  miles) 
per  second. 

7.  Show  that  if  a  meteor  falls  upon  a  planet  with  velocity 
equal  to  that  which  it  would  acquire  if  it  fell  from  rest  at  an 
infinite  distance  from  the  planet  under  the  planet's  attraction, 
its  kinetic  energy  will  be  proportional  to  the  product  of  the 
radius  of  the  planet  and  the  force  of  gravity  on  its  surface. 

8.  Given  that  a  falling  body  reaches  the  earth's  surface 
with  a  velocity  v0,  compute  the  height  through  which  it  has 
fallen  from  rest,  first,  on  the  assumption  that  the  force  which 
urged  it  was  constant,  and,  secondly,  on  the  assumption  that 
the  force  varied  inversely  as  the  square  of  the  distance  of  the 
body  from  the  earth's  centre,  and  prove  that  the  difference 
between  the  reciprocals  of  the  answers  you  obtain  is  equal  to 
the  reciprocal  of  the  earth's  radius. 

9.  Given  the  radius  of  the  earth  in  centimetres  (6.37  X  108), 
the  mass  of  the  earth  in  grammes  (6.14  X  1027),  the  radius  of 
the  sun  (6.97  X  1010),  the  mass  of  the  sun  (2.03  X  1033),  and 
the  mean  distance  between  the  centres  of  the  earth  and  sun 
(1.49  X  1013),  find  the  time  when  the  sun  and  earth  would  come 
together,  if  both  were  arrested  in  their  pafchs.     Prove  that  the 
acceleration  due  to  gravity  is  at  the  sun's  surface  about  27.6  g. 

10.  A  body  of  mass  m  falls  from  rest  near  the  surface  of 
the  earth  and  is  retarded  by  the  resistance  of  the  air,  which 
is  \v2  dynes  when  the  velocity  is  v.    Show  that  if  s  represents 
the  space  passed  over  up  to  the  time  t,  and  if  /x  =  X/m  and 
c*  =  g/n,2  ^t  =  log  [(c  +  v)  /(c-  v)^  2^s  =  log  [6'Y  (c* - i,2)], 
v2  =  c2(l  —  e"2**"),  and  ps  =  log  cosh  Qict). 

Show  that  if  the  body  were  thrown  upward  with  initial 
velocity  v0,  we  should  have  tan  (/xc£)  =  C(VQ  —  v)  /  (c2  -f  v0v). 

If  in  the  case  of  the  falling  body  v  is  the  actual  velocity 
and  v'  the  velocity  which  would  be  required  by  falling 
through  the  same  distance  in  vacua, 

v*/v<*  =  1  -  £  v'2/c2  +  J  v' 4/c*  -  ^v'6/c*  +  - . . . 


MISCELLANEOUS    PROBLEMS.  341 

11.  Show  that  the  periodic  time  of  a  planet  moving  about 
a  fixed  sun  of  mass   m  in  a  circular  orbit  of  radius  r  is 
2  TIT  3  /V  A:  HI,   where   1/&  is  the  ratio  of   the  absolute  unit 
of  force  in  the  given  length-mass-time  system  to  the  corre 
sponding  attraction  unit  ;    and,  assuming  that  the  diminu 
tion  of  gravity  at  the  equator  due  to  the  earth's  rotation  is 
about  ^^th  of  the  whole,  and  that  the  mean  distance  of  the 
moon  from  the  earth's  centre  is  about  60  times  the  earth's 
radius,  compute  the  length  of  the  month. 

12.  When  a  particle  moves  in  any  plane  curve,  the  tangen 
tial  and  interior  normal  acceleration  components  are  Dtv  and 
v2  1  p,  while  the  acceleration  components,  taken  along  and  per 
pendicular  to  the  radius  vector  which  joins  any  fixed  point  in 
the  plane  used  as  the  origin  of  a  system  of  polar  coordinates, 
to  the  particle,  are  D*r—r(Dtff)*  and  Dt(i*.DtG)/r  respec 
tively.     If   the    resultant    acceleration    is    always    directed 
towards  the  origin,  Dt(i*DtG)  =  0  and  r2-D(0  =  h,  so  that  the 
areas  of  the  sectors  swept  over  in  any  two  time  intervals  by 
the  radius  vector  are  to  each  other  as  the  lengths  of  the  inter 
vals  :  if  p  represents  the  perpendicular  let  fall  from  the  origin 
upon  the  tangent  to  the  path,  vp  =  rDt0  =  h. 

The  acceleration  towards  the  origin  is 


and,  if  u  represents  the  reciprocal  of  r,  this  may  be  written 

AV  (u  +  Dfu). 
Since  v2  =  h2  [V  +  (D9uf]  , 

J  Dt  (r)2  =  h*Dtn  (u  +  Dfu)  =  -  R  •  Dtr. 

In  the  case  of  a  planet  describing  a  plane  orbit  about  a 
fixed  primary  centred  at  the  origin 

R  =  nW  =  AV  (u  +  D*^  .   Qr  Dss  +  «  =  <), 

2  2 

where      z  =  u  —    ^  >  so  that  u  =    r2  +  C  sin  (6  —  X). 


342  MISCELLANEOUS    PROBLEMS. 

This  is  the  equation  of  a  conic  section  referred  to  a  focus  as 
origin  :  if  e  is  the  eccentricity  and  m  the  distance  of  the  focus 
from  the  directrix,  C  =  —  1  jm  and  A2  /ft2  =  am.  The  angle  \\i 
between  the  radius  vector,  drawn  from  the  origin  to  any  point 
on  the  orbit  and  the  tangent  at  the  point,  is  given  by  the 
equation,  ctn  \f/  =  —  r  •  C  •  cos  (0  —  A).  Assuming  that,  when  0 
is  zero,  \f/  =  a,  r  —  a,  and  v  =  v0,  show  that  h  =  v0a  •  sin  a,  and 
1  —  e2  =  (2  ^  —  v02a)  h2  /  an*.  Discuss  separately  the  three 
cases  where  v<?  is  respectively  less  than,  equal  to,  and  greater 
than  2  p2  /a,  and  find  the  lengths  of  the  semiaxes  of  the  orbit. 
Show  that,  if  a  =  90°  and  if  v*a  =  /*2,  the  orbit  will  be  circu 
lar;  show  also  that,  if  T  is  the  periodic  time  of  the  planet 
and  a  the  semiaxis  major  of  its  orbit,  /x2T2  =  47r2&3. 
13.  Assuming  that  the  equation 


*  (sin  i  a,  *), 

where  sin  <£  •  sin  J.  a  =  sin  ^  0,  and  a  is  the  angular  amplitude 
on  one  side  of  the  vertical,  gives  the  time  occupied  by  a 
simple  pendulum  of  length  a  in  going  from  the  vertical 
position  to  a  position  in  which  the  thread  makes  the  angle  6 
with  the  vertical  ;  and  that  the  complete  time  of  swing  is 

2  TT\~  [1  +  i  sin2  J  a  +  e\  sin4  1  a  +  •  •  •  ]  ; 
J 

assuming  also  that  a  rigid  body  swinging  about  a  horizontal 
axis  under  gravity  moves  like  a  simple  pendulum  of  length 
kz/h  where  h  is  the  distance  of  the  centre  of  gravity  from  the 
axis  and  k  is  the  radius  of  gyration  ;  show  how  a  pendulum 
may  be  used  to  measure  the  force  of  gravity  at  a  point. 

If  the  earth  were  a  homogeneous  sphere,  would  a  clock 
which  at  a  given  temperature  keeps  correct  time  on  the 
earth's  surface  lose  or  gain  at  the  same  temperature  at  the 
bottom  of  a  deep  mine  ?  Assuming  that  if  g^  and  gQ  are 
the  accelerations  due  to  gravity  at  sea  level,  in  latitude  X  and 


MISCELLANEOUS    PROBLEMS. 


343 


at  the  equator  respectively  y^  =  y0  (1 -f  .005226  sin2  A)  and 
y0  =  978.1 ;  show  that  the  lengths  of  the  seconds  pendulum 
at  the  north  pole,  in  latitude  45°,  and  at  the  equator,  are  about 
99.6  centimetres,  99.3  centimetres,  and  99.1  centimetres. 

A  pendulum  which  beats  seconds  on  the  earth's  surface 
gains  n  seconds  per  day  in  a  mine  h  metres  deep.  Show  that 
if  p0  is  the  mean  density  of  the  earth  and  p  the  density  of  the 
surface  stratum, 

2 )  approximately. 

Po  / 


86400 


14.  Assuming  that  the  earth  is  a  homogeneous  sphere,  of 
radius  6.37  X  108  centimetres  and  of 
mass  6.14  X  1027  grammes,  rotating 
uniformly  about  its  axis  in  86164 
seconds,  so  that  the  velocity  of  a  point 
on  the  equator  is  about  463  metres  per 
second,  show  that  the  angular  veloc 
ity  of  the  earth  is  0.00007292  or 
about  (13713)- l  radians  per  second, 
and  that  the  downward  acceleration  at 
the  equator  is  by  3.39  centimetres  per 

second,  or  about  — - ,  less  than  the  acceleration,  G,  at  the  poles. 
Show  also  (Fig.  122)  that  the  acceleration  of  gravity  towards 

the  earth's  centre  at  the  latitude  A  is  G  (  1  -      ^ —  V  the  devi- 

\          ^89  / 

/     .  \ 

ation  of  the  plumb  line  tan  -1  (  — — — ^L_  l    an(j  ^he  hori_ 

\289  —  cos2  Ay 

zontal  component  of  apparent  gravitation  — -  sin  A  cos  A. 


FIG.  122. 


15.  If  in  the  case  of  any  homogeneous  spherical  body 
rotating  uniformly  about  its  axis,  the  polar  gravity  accelera 
tion  and  the  equatorial  gravity  acceleration  be  gp  and  ge 


344  MISCELLANEOUS    PROBLEMS. 

respectively,  the  acceleration  of  gravity  towards  the  earth's 
centre  in  latitude  A  is  (g*p  sin2  A  -f  g\  cos2  A)  and  the  deviation 
of  the  plummet  from  the  geometrical  vertical  is 


tan  """       Sin  X  COS 


X  H-  ge  cos2  X 


} 


16.  A  bicycle  and  its  rider  weigh  together  75  kilogrammes. 
Show  that  if  the  machine  were  driven  first  eastward  and  then 
westward  in  this  latitude  at  a  velocity  of  10  metres  per  second, 
the  difference  between  the  pressures  on  the  ground  in  the 
two  cases  would  be  about  16.5  grammes. 

17.  The  centre  of  a  planet  of  radius  a  moves  around  a  sun  of 
mass  M  in  a  circular,  or  bit  of  radius  r.     Compute  the  pressures 
exerted  on  the  surface  of  the  planet  by  two  equal  particles,  each 
of  mass  m,  situated  respectively  on  the  points  of  the  planet 
nearest  and  farthest  from  the  sun.     Show  that  the  difference 
between  these  pressures  is  small  compared  with  the  difference 
between  the  attractions  of  the  sun  upon  these  particles. 

What  is  the  difference  between  the  apparent  weights  of  a 
body  of  mass  m  on  the  earth's  equator  about  September  21,  at 
noon  and  at  midnight  ? 

18.  Two  rods  AB  and  CD,  both  of  line  density  p,  are  placed 
parallel  to  each  other.     Show  that  the  force  on  either  in  the 
direction  of  its  length  is 

2/1      AC  +  AD  +  CD  BC+  BD+  CD} 

p   I    S  AC  +  AD  -CD         gBC  +  BD-CD)' 


The  component  of  the  mutual  attraction  perpendicular  to  the 
rods  is  2p2(BC  —  BD  —  AC  +  AD)/r,  where  r  is  the  perpen 
dicular  distance  between  them. 

19.  The  sides  of  a  triangle  are  formed  of  three  thin  uni 
form  rods  of  equal  density.  Prove  that  a  particle  attracted 
by  the  sides  is  in  equilibrium  if  placed  at  the  centre  of  the 
inscribed  circle.  [M.  T.] 


MISCELLANEOUS    PROBLEMS.  345 

20.  Every  particle  of  three  similar,  uniform  rods  of  infinite 
length  lying  in  the  same  plane,  attracts  with  a  force  varying 
inversely  as  the  square  of  the  distance  :  prove  that  a  particle 
subject  to  the  attraction  of  the  rods  will  be  in  equilibrium,  if 
it  be  placed  at  the  centre  of  gravity  of  the  triangle  enclosed 
by  the  rods.     [M.  T.] 

21.  The  attraction  of  the  straight  rod  AB  at  a  point  P 
is  the  resultant  of  two  forces,  each  equal  to  /,  acting  at  P 
towards  the  extremities  of  the  rod, 

where  f=2m-  AB/[(AP  +  BP)2  -  AB-~\. 

Find  the  value  of  /  when  P  lies  on  an  ellipse  the  foci  of  which 
are  the  extremities  of  the  rod.     [Routh.] 

22.  If  the  direction  at  the  point  0  of  the  attraction  of  every 
portion  of  a  uniform  plane  curvilinear  wire  bisects  the  angle 
subtended  at  0  by  that  portion,  the  wire  is  either  straight 
or  has  the  form  of  a  circumference  with  centre  at  0.    [Routh.] 

23.  If  the  law  of  attraction  be  the  inverse  square,  two 
curvilinear  rods  in  one  plane  exert  equal  attractions  at  the 
origin  if  the  densities  at  points  on  the  two  rods  on  any  radius 
vector  drawn  through  the  origin  are  proportional  to  the  per 
pendiculars  from  the  origin  on  the  tangents.     [Routh.] 

24.  Prove  directly  from  the  formula  for  the  attraction  of  a 
slender  straight  wire,  that  the  attraction  at  a  point  P,  due  to 
an  infinite  homogeneous  cylinder  of  any  form,  is  twice  that  of 
so  much  of  the  cylinder  as  is  cut  off  by  a  double  cone  formed 
by  the  revolution  about  a  line  through  P,  parallel  to  the 
generating  lines  of  the  cylinder,  of  a  line  which  cuts  this  line 
at  P  at  an  angle  of  60°. 

25.  A  uniform  wire  AB  in  the  form  of  a  circular  arc  has 
its  centre  at  0.     Prove  that  the  component  of  the  attraction, 
at  any  point  P,  in  a  direction  perpendicular  to  the   plane 
containing  P  and  the  normal  at  0  to  the  plane  of  the  arc,  is 
af*(>'i~l  ~~  rz~l) I h>  where  t\  =  AP,  r  =  BP,  h  is  the  projection 
of  OP  on  the  plane  of  the  arc,  and  p.  the  line  density  of  the  wire. 


346  MISCELLANEOUS    PROBLEMS. 

26.  Prove  that  the  attraction  in  the  direction  PO  at  a  point 
P  on  the  circumference  of  a  circle  the  centre  of  which  is  0, 
due  to  an  infinitely  long,  straight  filament  of  given  density 
passing  through  a  point  Q  in  the  circumference  and  perpen 
dicular  to  its  plane,  is  the  same  wherever  the  point  Q  is.     If 
the  filaments  of  a  homogeneous  columnar  distribution  of  given 
mass  per  unit  length  are  so  arranged  that  the  cross-section  is 
a  circle    passing   through  a  point  P,  the   attraction  of  the 
distribution  on  P  will  be  a  maximum.     [Tarleton.] 

27.  A  water  tower  in  the  shape  of  a  cylinder  of  revolution 
is  100  feet  high  and  10  feet  in  diameter.     The  mass  of  the 
tower  and  contents  is  8400  pounds  per  foot  of  height.     With 
out  the  help  of  pencil  or  paper,  guess,  to  within  one  per 
cent  of  the  truth,  the  value  in  f.p.s.  attraction  units  of  the 
horizontal  component  of  the  attraction  due  to  the  tower  at  a 
point  at  its  foot  just  outside  it. 

28.  Prove  that  at  a  point  on  the  edge  of  an  infinite  homo 
geneous  cylinder  of  semicircular  cross-section,  the  components 
of  the  attraction  across  the  plane  face  perpendicular  to  the  axis, 
and  normal  to  the  face,  are  irakp  and  2  akp  respectively,  and 

show  that  gravity  is  diminished  by  the  fraction  —  • —  at 

the  middle  of  the  surface  of  a  long  straight  canal  of  semi 
circular  section,  a  being  the  radius  of  the  semicircle,  r  the 
radius  of  the  spherical  earth,  p'  the  density  of  water,  p  that 
of  the  surface  stratum  of  the  earth,  and  p0  the  earth's  mean 
density.  The  corresponding  quantity  in  the  case  of  a  canal 
of  rectangular  cross-section  of  depth  a  and  breadth  2  a  is 

TT  +  2  •  log  2   3_a  p-  p' 
TT  4r       po 

29.  An  infinitely  long  homogeneous  prism  has  a  rectangular 
cross-section  of  length  a  and  breadth  ft.     Assuming  that 


flog  (a 


=  x-  log  (a2  +  x2)  -  2  x  +  2  a  tan~  l  (x  /  a), 


MISCELLANEOUS    PROBLEMS.  347 

show  that  at  any  point  on  one  of  the  edges  the  components  of 
the  attraction  along  the  sides  a  and  b  of  the  cross-section 
through  the  point  are 

kp  \2  a  tan-1  (6  /a)  +  b  •  log[(a2  +  b'2)  /b2^ 
and  kp\2b  tan-1  (a/b)  +  a  .  log  [(a2  +  b2)  /a2]  \. 

If  the  ratio  of  b  to  a  is  large,  the  first  of  these  quantities  is 
nearly  equal  to  irapk.  Show  that  the  apparent  latitude  of  a 
point  on  one  edge  of  a  long,  deep,  narrow  crevasse  of  breadth 
a,  running  east  and  west,  is  altered  by  the  angle  3pa/4p0r, 
nearly,  by  the  presence  of  the  crevasse.  [Thomson  and  Tait.] 
30.  Assuming  that  the  attraction  of  a  homogeneous  cylinder 
of  revolution,  of  density  />,  radius  a,  and  height  h,  upon  a  unit 
particle  at  the  centre  of  one  of  its  ends,  is 

a         1'1     "S       1'1'3     "5 


h        1-1    7/3      1-1-3 
or 


according  as  a  is  small  or  large  compared  with  h,  and  con 
sidering  that  the  mean  surface  density  of  the  earth  is  3  times 
and  the  mean  density  of  the  whole  earth  5.5  times  the  density 

of  sea  water,  obtain  Siemens's  expression,  ?  for  the  dimi 

nution  of  gravity  at  a  point  on  the  ocean  where  the  depth  is  h. 
Is  the  intensity  of  gravity  at  the  centre  of  the  mouth  of  a  ver 
tical  mine  shaft  20  feet  in  diameter  appreciably  less  than 
before  the  shaft  was  dug  ?  Show  that  if  h  =  a,  the  attraction 
due  to  a  cylinder  of  revolution,  at  the  centre  of  one  of  its 
ends,  is  2  trkpa  (2  —  V2).  The  attraction  due  to  the  earth 


.,2 


at  a  point  P  at  a  height  h  above  the  surface,  is  — -^>  or 

(r  -j-  A) 

g  (  1  -      -  J    approximately,    where  r  is   the    radius   of   the 
earth.     If  p0  is  the  earth's  mean  density,  g  =  $  vkp0r.    If  P  is 


348  MISCELLANEOUS    PROBLEMS. 

at  the  centre  of  a  wide  plateau  of  height  li  made  of  matter  of 
density  p,  the  additional  attraction  due  to  the  plateau  is  about 
2  irkph,  or  3  yph/2  pQr,  so  that  if  p  =  ^  p0,  the  whole  attraction 
5  h\ 

/ 

31.  A  vertical  solid  cylinder  of  height  a  and  radius  r  is 
divided  into  two  parts  by  a  plane  through  the  axis.  Show 
that  the  resultant  horizontal  attraction  of  either  part  at  the 
centre  of  the  base  is 

v  +  V^T^ 


is  nearly  fMw—  1 


32.  A  right  circular  cylinder  is  of  infinite  length  in  one 
direction  and  is  homogeneous.  Prove  that  if  the  finite  extrem 
ity  be  cut  off  perpendicularly  to  the  generators,  the  attraction 

on  a  unit  particle  placed  at  the  centre  of  this  end  is  -     —  > 

where  M  is  the  mass  per  unit  of  length.  If  the  cylinder  be 
elliptic,  of  the  same  density  and  mass  per  unit  of  length  as 
before,  and  of  eccentricity  e,  then  the  attraction  will  be  n 
times  the  former  value,  where 

dO 


Vl  -  e2sin20 

33.  A  homogeneous,  right  circular  cylinder  of  density  p 
stands  on  the  plane  z  =  0,   and  is   infinite  in  the  positive 
direction  of  the  axis  of  z.     Show  that  the  z  component  of 
its  attraction  at  a  point  P  of  its  base  is  kpl,  where  I  is  the 
perimeter  of  an  ellipse  having  the  base  for  the  auxiliary  circle 
and  P  for  one  focus. 

34.  Show  that  the  attraction  at  any  outside  point  P,  due 
to  a  uniform  plane  lamina  of  any  shape,  yields  a  component 
normal  to  the  lamina,  equal  to  the  product  of  the  solid  angle 
subtended  at  P  by  the  lamina,  and  a  quantity  which  does 
not  depend  upon  P's  position. 


MISCELLANEOUS    PROBLEMS.  o4i"4 

35.  Show  that  the  component  perpendicular  to  its  axis,  of 
the  flM"""*5""  of  a  thin,  homogeneous,  circular,  cylindrical 
sheet  of  height  2k  and  radius  a,  has  at  any  point  on  one 
of  the  circular  bounding  odges  of  the  cylinder  the  value 


where  ^= 


9  .   M 


36.  An  infinitely  long  plane  sheet  of  constant  width  has 
a  ff»a11  thickness  &  and  is  made  of  homogeneous  matter  of 
density  p.     This  strip  cuts  a  plane  perpendicular  to  its  long 
edges  in  the  line  AB  :  show  that  the  attraction  of  the  strip  at 
any  point  P  in  this  plane  has  a  component  2  kp&  log  (PB/PA) 
parallel  to  AB,  and  a  component  2  fyS  -  Z  APB  perpendicular 
toAB. 

37.  Every  diameter  of  a  certain  circle  subtends  a  plane 
angle  2$  at  a  ffrrtam  point  P  on  the  axis  of  the  circle  ;  show 
that  the  circle  subtends  at  P  the  solid  angle  2«-(l  —  cos  0). 

38.  Compare  the  attractions,  at  the  vertex  of  a.  homoge 
neous  oblique  cone  which  has  a  plane  base,  due  to  the  whole 
cone  and  to  so  much  of  it  as  lies  between  the  vertex  and  a 
plane  which  bisects  at  right  angles  the  perpendicular  drawn 
from  the  vertex  to  the  base. 

39.  Prove  the  truth  of  the  theorem  which  Xewton  states 
in  the  f  ollowing  words  :  «  Si  corporis  attracti.  ubi  attrahenti 
contiguum  est,  attractio   longe   fortior  est,   quam   cum   vel 
minimo  intervallo  separantur  ab  invicem  :  Tires  particularum 
trahentis  in  i  liuiiiiu.  eorporis  aHiauli,  degtesflunt  in  ratione 
plusquam  dnptJcita  distantianim  a  particulis.     Si  particula 
rum,  ex  quibus  corpus  attracti  vam  componitur.  Tins  in  reoessu 
corporis  attracti  decrescunt  in  triplicata.  vel  plusquam  tripli- 
cata  ratione  distantiarum  a  particulis,  attractio  longe  fortior 
erit  in  contactu,  quam  cum  trahens  et  attractum  interrallo  vel 
••••••^  separantur  ab  invicem."      [PAiZ  JVfi/,  Prime. 

Sectio 


350  MISCELLANEOUS    PROBLEMS. 

40.  Two  homogeneous  solids  made  of  the  same  material 
are  bounded  by  similar  surfaces.     Show  that  the  intensities 
of  their  attractions  at  two  points  similarly  situated  respec 
tively  with  regard  to  them,  are  in  the  ratio  of  the  correspond 
ing  linear  dimensions  of  the  solids.     Hence  prove  that  the 
attractions  at  points  on  a  given  diameter  inside  a  solid  homo 
geneous  ellipsoid  are  proportional  to  the  distances  of  these 
points  from  the  centre. 

41.  Prove  that  the  attraction,  at  very  distant  points,  of  any 
system  which  has  an  axis  of  symmetry,  may  be  represented  as 
emanating  from  two  equal  poles  of  the  same  sign  situated  on 
the  axis. 

42.  Show  that  the  component,  at  the  origin,  in  the  direc 
tion  of  the  x  axis,  of  a  given  particle  m,  is  the  same  wherever 
on  the  surface  m  •  cos  (x,  r)/r*  =  c,  where  c  is  a  given  constant, 
the  particle  lies.     If  it  is  anywhere  without  the  surface,  the 
component  will  be  less   than  if   it  were  anywhere  within. 
Hence  prove  that  the  attraction  of  a  given  mass  M  for  a  point 
on  its  surface  will  be  greatest  if  the  boundary  of  M,  referred 
to  the  given  point,  is  a  surface  of  the  family  cos  0  =  \-  r*. 

43.  If  the  earth  be  considered  as  a  homogeneous  sphere  of 
radius  r,  and  if  the  force  of  gravity  at  its  surface  be  y,  show 
that  from  a  point  without  the  earth,  at  which  the  attraction  is 


the  area  2  Trr2  [  1  —  \l-  -  -  }  on  the  surface  of  the 


n 
earth  will  be  visible. 

44.  The  laws  of  attraction  for  which  the  attraction  of  a 
homogeneous  shell  on  any  external  particle  is  the  same  as  if 
the  shell  were  concentrated  at  its  centre,  are  the  "  law  of  the 
inverse  square  "  and  the  "  law  of  the  direct  distance." 

45.  Whatever  may  be  the  law  of  attraction,  the  intensity 
of  the  force  exerted  by  the  smaller  of  two  concentric  solid 
homogeneous  spheres  at  any  point  on  the  surface  of  the  larger, 
is  to  the  intensity  of  the  force  exerted  by  the  larger  at  any 


MISCELLANEOUS    PROBLEMS.  351 

point  on  the  surface  of  the  smaller,  in  the  ratio  of  the  square 
of  the  radius  of  the  smaller  to  the  square  of  the  radius  of  the 
larger.  [Minchin.] 

46.  Prove  that  if  /  be  an  external  point  and  C  the  centre  of 
a  sphere,  the  sphere  on  1C  as  diameter,  the  sphere  with  centre 
/  and  radius  1C,  or  the  polar  plane  of  /,  will  divide  the  sphere 
into  two  parts  which  exert  equal  attractions  at  /,  according 
as  the  law  of  attraction  is  the  inverse  square,  the  inverse  cube, 
or  the  inverse  fourth  power  of  the   distance.     [St.  John's 
College.] 

47.  Two  sectors  are  cut  from  a  homogeneous  shell  bounded 
by  two  concentric  spherical  surfaces  of  radii  i\  and  r2,  by  a 
conical  surface  of  revolution  of  half  angle  0  and  with  vertex 
at  the  centre  0  of  the  shell.     The  attractions  at  a  point  P 
without  the  shell  on  the  axis  of  the  cone,  on  its  inner  side, 
at  a  distance  c  from  0,  due  to  the  portions  of  the  shell  which 
lie  respectively  without  and  within  the  cone  are  Fl  and  Pz. 
Show  that  jp\  is  equal  to  the  difference  between  the  values 
when  r  =  rz  and  >•  =  )\  of  a  quantity  A,  and  that  F2  is  equal 
to   the   difference   between    the    corresponding   values   of   a 
quantity  B  where 

rt    7 

A  =  —^  [J  r8  -  oii  (i  r2  -  f  c2  +  c2  cos20  +  i  re  cos  6) 

4-  c3  cos  0  sin2  0  •  log  (o>*  +  r  —  c  cos  0)] , 

B2  =  ^jf£  [i  r3  +  ^  (J  ,*  -  f  c2  +  c2  cos20  +  J  w  cos  6) 

—  c3  cos  B  sin2  0  -  log  (o>*  +  >•  —  c  cos  0)] , 

and  o>  =  c2  +  ?*2  —  2  cr  cos  0. 

The  attractions  of  the  halves  of  the  shell  farthest  from  P 
and  nearest  to  it  are 


352  MISCELLANEOUS    PROBLEMS, 

and 


respectively.     If  the  mass  of  the  whole  shell  is  M  and  if  the 
shell  is  thin,  the  attractions  at  P  due  to  the  sectors  are 

kM  (  ^       r  —  c  cos  0\        ,  kM  f        r  —  c  cos  0\ 

~f\      o    I     -^-   ""I 


2c2  V  PL      )          2c2  \  PL      ) 

where  L  is  any  point  on  the  common  rim  of  the  sectors. 

48.  Prove  that  the  attraction  due  to  a  homogeneous  hemi 
sphere  of  radius  r  is  zero  at  a  point  in  the  axis  of  the  hemi 
sphere  distant  f  r  approximately  from  the  centre  of  the  base. 

49.  A  segment  of  height  h,  cut  from  a  homogeneous  sphere 
of  density  p  and  radius  a  by  a  plane  distant  a  —  h  from  the 
centre  of  the  sphere,  attracts  a  unit  particle  on  the  axis  of 
the  segment  at  a  distance  b,  greater  than  the  radius,  from 
the  centre  of  the  sphere,  with  a  force 

27T&J  h  +         *         |(2  c2  +  3  ac)c  -  (2  c2  +  3  ac  +  ah  +  ch) 
|_         o  (c  -\-  (ij    L 

Vc2  -f  2  ch  +  2  ah  |    ,  where  c  =  b  —  a. 

If  c  =  0,  this  becomes  2>n-kph  4  1  —  -  \  — -  r  '    Assuming  this 

to  be  true,  show  that  the  attraction  of  a  homogeneous  hemi 
sphere  upon  a  particle  at  its  vertex  is  to  the  attraction  of  the 
circumscribing  cylinder  of  the  same  density  as  529  to  586, 
nearly.  Show  that  the  attraction,  at  its  vertex,  of  a  slice 
2  miles  thick  cut  from  the  earth,  and  the  attraction  of 
an  infinite  disc  of  the  same  thickness  and  density  upon  a 
point  at  the  centre  of  one  of  its  'faces,  differ  by  about  one 
per  cent  of  either. 

50.  Show  that  if  the  earth  were  made  up  of  two  homogeneous 
solid  hemispheres  bf  densities  p  and  p'  separated  by  the  plane 


MISCELLANEOUS    PROBLEMS.  353 

of  the  equator,  the  deviation  of  the  plumb  line  from  the  zenith 
at^  any  point  of  the  equator  would  be  tan"1  (  -  • — f  V 

51.  Show  that  the  attraction  at  the  origin  due  to  the  homo 
geneous  solid  bounded  by  the  surface  obtained  by  revolving 
one  loop  of  the  curve  r2  —  a2  •  cos  2  6,  is  ^  irakp. 

52.  A  mountain  of  the  form  of  a  surface  of  revolution  with 
vertical  axis  and  elliptic  outline  stands  on  a  horizontal  plane 
which  contains  the  centre  of  the  ellipse.     Find  the  horizontal 
component  of  its  attraction  at  a  point  of  the  base.     Show  that 
if  the  mountain  is  2  miles  high  and  4  miles  broad  at  the  base, 
and  if  the  density  of  the  mountain  and  of  all  the  matter  in  its 
neighborhood  is  half  the  mean  density  of  the  earth,  the  plumb 
lines  close  to  its  base  on  the  north  and  south  sides  will  make 
with  each  other  an  angle  greater  by  about  51  seconds  of  arc 
than  the  corresponding  difference  of  geocentric  latitude. 

53.  The  attraction  at  the  point  (0,  0,  —  b)  of  so  much  of  the 
homogeneous  paraboloid   x2  -f-  if  =  \z   as   lies   between   the 
planes  z  =  0,  z  =  h  is 


\h  -  V(6  +  A)2  +  h\  +  b  -  H-log(2&  +  *  A) 


+  £  A  •  iog(V(7»  +  hy+ h\  +  b  +  h  - 

54.  If  a  body  M  be  divided  into  two  rigid  portions,  A  and 
B,  the  resultant  action  of  each  portion  upon  itself  is  nil,  and 
the  attraction  between  A  and  B  is  the  same  mathematically 
as  the  attraction  between  M  and  B.     To  find,  therefore,  the 
attraction  between  two  equal  homogeneous   hemispheres  so 
placed  as  to  form  a  sphere,  we  may  integrate  through  either 
hemisphere  the  product  of  the  density  and  the  component 
normal  to  the  flat  face  of  the  hemisphere,  of  the  attraction  due 
to  the  whole  sphere.     Show  that  the  result  is  3  kM2/16  a2. 

55.  Show  that  the  resultant  attraction  between  the  two 
parts  into  which  a  homogeneous  sphere  is  divided  by  a  plane 


354  MISCELLANEOUS    PROBLEMS. 

is  equal  to  the  mass  of  either  part  multiplied  by  the  intensity 
of  gravitation  at  its  centre  of  mass. 

56.  Prove  that  the  pressure   per  unit  of  length  on   any 
normal  section  of  a  spherical  shell  of  mass  M  and  radius  a, 
due  to  the  mutual  gravitation  of  the  particles,  tends  to  the 
limit  7cJf2/167r&3,  as  the  thickness  of  the  shell  is  indefinitely 
diminished.     [M.  T.] 

The  mass  of  the  unit  length  of  an  infinite  homogeneous 
cylinder  of  revolution  of  radius  a  which  is  divided  into  two 
parts  by  a  plane  through  its  axis  is  M.  Show  that  the  pres 
sure  between  the  two  parts  due  to  their  mutual  attractions  is 
4  kM2/3  Ira  per  unit  length  of  the  cylinder. 

57.  If  R  and  S  denote  the  components  of  attraction  of  a 
gravitating  system  symmetrical  with  respect  to  a  straight 
axis,  taken  along  and  perpendicular  to  the  axis,  then 


where  r  and  z  are  columnar  coordinates.  [St.  John's  College.] 
58.  If  the  point  of  application  of  a  force  F  move  by  the 
path  s  from  the  point  A  to  the  point  B,  the  force  is  said  to 
do  work  during  the  journey,  equal  in  amount  to  the  line  inte 
gral  taken  along  s  of  the  tangential  component  of  F.  If  the 
components  of  F  parallel  to  the  coordinate  axes  are  X,  Y,  Z, 
and  if  dx,  dy,  dz  are  the  projections  on  these  axes  of  an 
element  ds  of  the  path,  we  have  the  expressions 


W=    C    F.  cos(s,F)ds 

J  A 

XB 
F  [cos  (x,  s)  •  cos  (x,  F) 
__ 


__ 
+  cos  (y,  s)  •  cos  (y,  F}+  cos  («,  s)  •  cos  (z,  F)~\  ds 

CE 
—    I     \X-  cos  (x,  s)  -f  Y-  cos  (y,  s)+  Z-  cos  («,  s)]  ds 

*/  A 

=    CB  Xdx  +  Ydy  +  Zdz. 

J  A 


MISCELLANEOUS    PROBLEMS.  355 

If  a  function  O  exists  such  that 

X=Dxtt,  r=Z>yO,  Z=DZQ  •    W=   C*  dV,  =  £lB-£lA: 

J  A 

such  a  function  is  called  a  potential  function  or  a  force 
function  of  the  given  force.  The  work  done  by  a  force  which 
has  a  potential  function,  when  its  point  of  application  moves 
completely  around  any  closed  path,  is  zero,  and  such  a  force 
is  said  to  be  conservative.  The  work  done  by  a  conservative 
force  as  its  point  of  application  moves  from  A  to  B  is  inde 
pendent  of  the  path  s. 

Prove  by  actual  integration  along  the  different  paths,  that  the 
work  done  by  the  force  X=  3  x2  +  2  y,  Y  =  4  y3  +  2ar,  Z  =  0, 
when  its  point  of  application  moves  from  the  origin  to  the 
point  (2,  2,  0),  is  32,  whether  the  path  be  a  straight  line,  or 
the  parabola  if  =  2  x  in  the  xy  plane,  or  a  combination  of  a 
straight  line  from  the  origin  to  (2,  0,  0)  and  another  straight 
line  from  this  point  to  (2,  2,  0).  Show  that  the  derivative 
with  respect  to  x  of  any  function  of  the  form  xs  +  2  xy  +/(?/), 
where  f  is  arbitrary,  will  yield  X,  and  that,  by  a  proper 
choice  of/,  the  derivative  with  respect  to  y  can  be  made  equal 
to  Y'j  so  that  a  force  function  exists.  Prove  by  actual  inte 
gration  along  the  paths  that  the  work  done  by  the  force 


as  its  point  of  application  moves  from  the  origin  to  (2,  2,  0), 
is  not  independent  of  the  path.  In  this  case  no  potential 
function  exists,  since  it  is  impossible  to  give  such  a  form  to 
/,  in  the  general  expression  [z3  +  2  xy  +/(?/)],  which  has  X 
for  its  partial  derivative  with  respect  to  x,  that  the  partial 
derivative  of  the  expression  with  respect  to  y  shall  be  Y. 

Since  the  order  of  successive  partial  differentiations  of  any 
analytic  function  is  immaterial, 


or          DyZ  =  D2  F,  DZX  =  DXZ,  DXY=  DyX. 


356  MISCELLANEOUS   PROBLEMS. 

Show  that  this  necessary  condition  for  the  existence  of  a 
force  function  is  also  a  sufficient  one. 

59.  Prove  that  if  we  have  matter  attracted  to  any  number 
of  fixed  centres  with  forces  proportional  to  any  function  of 
the  distance,  or  if  we  have  matter  every  particle  of  which 
attracts  every  other  particle  according  to  any  function  of  the 
distance  between  the  particles,  there  exists  a  potential  func 
tion  the  derivative  of  which  in  any  direction  at  any  point 
gives  the  intensity  of  the  force  which  would  solicit  a  unit 
quantity  of  matter  concentrated  at  the  point  to  move  in  the 
given  direction. 

60.  If  r  represents  the  distance  of  any  point  Q  on  a  sur 
face  S  from  a  fixed  point  P,  and  if  a  is  the  angle  between  PQ 
and  the  normal  to  S  at  Q,  drawn  always  from  the  same  side 

/OOS  n 
— 3—  dS,  taken  over  any  portion  of  the  sur 
face,  gives  in  absolute  value  the  solid  angle  subtended  at  P 
by  this  portion,  and,  in  the  case  of  a  closed  surface,  this  value 
is  4  TT,  2  TT,  or  0,  according  as  P  is  within,  on,  or  without  S. 
Prove  that  the  volume  of  the  solid  enclosed  by  any  surface  S 

is  the  absolute  value  of  %  IT  cos  a  dS  taken  over  the  surface, 
whether  P  is  within  or  without  S.  Show  that  it  is  possible 
to  find  an  analogous  expression,  %  (r cos  ads,  for  the  area 

enclosed  by  a  plane  curve,  and  explain  in  this  case  the 
notation. 

61.  Show  that  the  absolute  value  of  the  component  parallel 
to  the  axis  of  x,  of  the  force  at  a  point  P,  within  or  without  a 
homogeneous  solid  body  of  any  form,  due  to  the  attraction  of 

.,:.,*              /•  cos  (x,n)-dS,  .  .,-  i 

this  body,  is  p  I *-* — " ,  where  n  is  an  interior  normal, 

taken  all  over  the  bounding  surface ;  and  prove  that  the 
component  parallel  to  the  axis  of  x  of  the  force,  at  a  point 
P,  due  to  the  attraction  of  a  homogeneous  infinite  cylinder 


MISCELLANEOUS    PROBLEMS.  357 

with  generating  lines  parallel  to  the  axis  of  z,  is  of  the  form 
2  /JL  I  cos  (x,  n)  •  log  r  •  ds,  where  the  integral  is  to  be  extended 

around  the  contour  of  the  section  of  the  cylinder  made  by  a 
plane  through  P  perpendicular  to  the  axis  of  z. 

62.  The  space  within  a  closed  surface  S  is  filled  with  homo 
geneous  matter  of  density  p.     Prove  that  the  value  at  the 
point  P,  of  the  potential  function  due  to  the  distribution,  is 

£  p  \  cosadS,  where  a  is  the  angle  which  the  normal  to  the  sur 
face,  drawn  inward  at  any  point  Q  on  it,  makes  with  QP. 

63.  Two  distributions  of  gravitating  matter  possess  a  com 
mon  closed   equipotential   surface.      Prove   that   if   all   the 
matter   of    both   distributions    be    within    this    surface,    the 
potentials  at  the  surface  due  to  the  two  distributions  are  to 
each  other  as  the  masses. 

64.  Prove  that  if  two  different  bodies  have  the  same  level 
surfaces  throughout  any  empty  space,  their  potential  func 
tions  throughout  that  space  are  connected  by  a  linear  relation. 
That  the  level  surfaces  should  be  the  same,  it  is  only  neces 
sary  that  the  resultant  forces  due  to  the  two  bodies  should 
coincide  in  direction. 

65.  Show   that   if   two    distributions    of   matter    have    in 
common    an    equipotential    surface    which    surrounds    them 
both,   all  their  equipotential    surfaces   outside  this  will   be 
common. 

66.  Show  that  if  we  have  matter  every  particle  of  which 
attracts  every  other  particle  with  a  force  proportional  to  the 
nth  power  of  the  distance,  the  attraction  at  any  point  within 
a   quantity   of   the    matter   will    be    infinite    if    n  +  2  <  0. 
[Minchin.] 

67.  Show  that  if  u,  v,  and  w  are  any  three  solutions  of 
Laplace's  Equation, 

V2  (iivw)  =  u  •  v2  (yw)  -f  v  •  v2  (?/«?)  +  w  •  y2  (uv). 


358  MISCELLANEOUS    PROBLEMS. 

68.  Show  that  the  potential  function  due  to  a  solid  hemi 
sphere  of  radius  a  and  density  p,  at  an  external  point  P 
situated  on  the  axis  at  a  distance  £  from  the  centre,  is 


the  upper  or  lower  sign  being  taken  according  as  P  is  on  the 
convex  or  plane  side  of  the  body. 

69.  A  sphere  with  centre  at  the  origin  has  a  radius  r  and  a 
density  given  by  the  law  p  =  ax  +  by  +  cz.  Prove  that  the 
value  at  any  external  point  (aj,  y,  z),  at  a  distance  It  from 
the  origin,  of  the  potential  function  due  to  the  sphere,  is 


70.  An  infinite  cylinder  of  radius  a  has  a  cylindrical  cavity 
of  radius  b  cut  out  of  it.      The  axes  of  the  cylinders  are 
parallel  but  not  coincident,  and  the  surfaces  do  not  intersect. 
Show  that  the  equipotential  surfaces  are  cylinders  the  equa 
tions  of  which  are  : 

(i)  ru2  —  rb2  =  Ci  within  the  cavity  ; 
(ii)  ra2  -2b*  log  ^  =  C2  within  the  mass  ; 
(iii)  a2  log  (  —  J  —  62  log  f  ~  J  =  C3  in  outside  space  ; 

where  ra  and  rb  are  the  distances  from  the  axes  of  the  cylinder 
and  cavity  respectively. 

71.  From  a  homogeneous  sphere  of  density  p  and  radius  a 
is  cut  an  eccentric  spherical  cavity  of  radius  b.    The  distances 
of  any  point  P  from  the  centre  of  the  sphere  and  the  centre 
of  the  cavity  are  rx  and  r2  respectively.     Show  that  VP,  the 
value  of  the  potential  function  at  P,  is  given  by  the  first, 
second,  or  third  of  the  subjoined  equations, 

V  +  27rbz  - 


MISCELLANEOUS  PROBLEMS.  359 

,  2&3    Q/  ,     rr\ 

rf  +  -  -  -  3  f  a-  -  — £-  ) , 
rz  \          2-irpJ 

Qi7                («*      l>*\ 
3  !>  P  =  4  Trp  ( )  , 

\ri      'V 

according  as  P  is  within  the  cavity,  within  the  mass,  or  with 
out  the  mass.  Indicate  by  a  rough  drawing  the  form  of  a 
line  of  force  within  the  cavity. 

72.  Show  that  the  lines  of  force  due  to  a  uniform  straight 
rod  are  hyperbolas  which  have  the  ends  of  the  rod  for  foci. 

73.  Show  that  formula  [59]  might  be  written 

VP  =  IL-  log  (ctn  £  PBA  -  ctn  £  PAS). 

74.  A  number  (n)  of  equal,  infinitely  long,  homogeneous, 
straight  filaments,  all  parallel  to  each  other,  cut  the  xy  plane 
normally  in  points  which  lie  uniformly  distributed  on  a  cir 
cumference  of  radius  a  with  centre  at  the  origin.     One  of 
these  points  is  at  the  point  (a,  0).    Show  that  the  value  of  the 
potential  function  at  the  point  (r,  0}  is 

m  •  log (;*2n  —  2  aV  cos  ?iO  +  a2"). 

75.  If  the  law  of  attraction  were  that  of  the  inverse  wth 
power  of  the  distance,  we  should  have 


If  the  density  had  the  same  sign  throughout  a  distribution  of 
matter,  the  potential  function  could  not  be  constant  in  any 
region  of  empty  space  unless  n  were  equal  to  2. 

76.  In  the  case  of  matter  every  particle  of  which  attracts 
every  other  particle  with  a  force  proportional  to  the  product 
of  their  masses  and  a  function  (/)  of  the  distance,  we  have 

V2FEE  f  f  f  [2/(?')/?'+/'(r)]pf/T.  Show  that  F  cannot 
satisfy  Laplace's  Equation  unless /(r)  =  K/r*. 


360  MISCELLANEOUS    PROBLEMS. 

77.  If  instead  of  the  polar  coordinates  r,  0,  <£,  the  independ 
ent  variables  are  r,  ^  </>,  where  /w,  =  cos  9,  Poisson's  Equation 
becomes 


Dr  (r2  -  Dr  F)  +  I)*  [(1  -  /*2)  DMF]  +  D\  V/  (1  -  /x2)  =  -  4  Trpr2. 

78.  If  instead  of  the  spherical  coordinates  r,  0,  <£,  the  coor 
dinates  ut  w,  </>  be  used,  where  u  =  1/r,  and  w  =  log  tan  £  0, 
Laplace's  Equation  becomes 

sin2  (2  tan-  1  ew)  u2  .  DJ>  V  +  iy  F  +  DJ  V  =  0. 

79.  Show  that  if  matter  be  distributed  symmetrically  about 
an  axis,  and  if  4  a,  Aa'  be  the  latera  recta  of  the  two  confocal 
parabolas,  with  this  line  as  axis,  which  meet  at  any  point, 
Laplace's  Equation  may  be  written  in  the  form 


80.  Prove  that  at  the  surface  of  an  attracting  body,  Z>X2F, 
Dyz  V,  Dz2  V  are  discontinuous  in  such  a  manner  that  if  n  repre 
sents  an  interior  normal  drawn  to  the  surface,  the  values  of  these 
quantities  at  any  point  just  within  the  attracting  mass  are 
smaller  than  at  a  neighboring  point  just    without,  by  the 
quantities    4  TT/J  cos2  (x,  ri),    4  irp  cos2  (y,  n),    4  -n-p  cos2  (z,   n), 
respectively. 

81.  A  portion  of  a  spherical  surface  is  occupied  by  a  thin 
shell  of  matter  of  uniform  density  <r,  which  attracts  according 
to  the  Newtonian  Law.     Prove  that  the  value,  at  any  point  on 
the  remaining  portion  of  the  surface,  of  the  potential  function 
due  to  this  distribution  of  matter,  is  a  a-  <o,  where  a  is  the 
diameter  of  the  sphere  and  w  the  solid  angle  subtended  at  the 
point  by  the  contour  of  the  portion  of  the  surface  occupied 
by  matter. 

82.  Show  that  in  so  far  as  a  transformation  from  one  set  of 
rectangular  axes  to  another  is  concerned,  D*V+  D*V  '+  DZZV 
and  (DxVy+(DyVy  +  (DZV)2  are  differential  invariants. 


MISCELLANEOUS    PROBLEMS.  361 

83.  The  potential  function  at  all  points  external  to  the 
sPhere  x2  +  if  +  *  =  a* 

a5  (ax2  +  ft/2  +  yz2  +  2  a'ljz  +  2  /fe  +  2  y'x^/r5. 

Show  that  if  there  be  no  matter  in  this  region,  a,  ft  and  y  must 
satisfy  a  certain  relation.  Show  that  if  inside  the  sphere  the 
density  be  uniform,  the  value  there  of  the  potential  function 

Wl11  be  c  +  XJP*  +  M*  +  vz2  +  2  a'yz  +  2  ft'zx  +  2  y'xy, 

where  c,  A,  p,  and  v  are  known.  Find  the  condition  that  under 
these  circumstances  the  equipotential  surfaces  inside  the 

x2      i/2      z2 
sphere  should  be  ellipsoids  similar  to  —  +  ^  +  ^  =  1- 

x  ^ 

84.  Prove  that  if  C<j>  (r)  •  dr  =  x  (>')  and  A--  x(>')  ^^  =  «A  (r)> 

r  r 

and  if  <#>,  x»  and  ^  vanish  at  infinity  and  are  finite  for  finite 
values  of  r  ;  mm'  x  (r)  represents  (1)  the  work  done  under  an 
attracting  force  mm'  <f>  (?•)  in  bringing  a  particle  of  mass  m' 
from  infinity  to  a  point  distant  r  from  another  mass  m  ; 
(2)  the  component,  parallel  to  the  rod,  of  the  attraction  of  a 
particle  m  on  a  straight  slender  rod  of  line  density  m',  if  the 
end  of  the  rod  is  at  a  distance  r  from  m  and  thr  other  end  at 
infinity.  Show  also  that  2  irarm  •  ty  (z)  repreient^  (1)  the  work 
done  in  bringing  from  infinity  to  a  point  distant  z  from  a  thin 
lamina  of  surface  density  o-,  a  particle  of  marfs  m\  (2)  the 
attraction  of  a  particle  m,  placed  at  a  distance  z  from  the  plane 
surface  of  an  infinite  solid  of  constant  density  o-. 

85.  Show  that  if  s  represents  a  direction  which  makes  the 
angles  a,  ft  y  with  the  coordinate  axes, 


+  D2  Fcos2  y  +  2  DxDyVcos  a  cos  (3 

+  2  DyDz  Fcos  (3  cosy  +  2  DJ)Z  Fcos  y  cos  a. 


362  MISCELLANEOUS   PROBLEMS. 

86.  When  the  line  of  action  of  the  attraction  of  a  body  at 
every  point  of  external  space  passes  through  a  point  0  fixed 
in  the  body,  the  body  is  said  to  be  centrobaric  and  0  is  called 
the  baric  centre.     The  lines  of  force  in  external  space  are 
straight  lines  passing  through  0,  and  the  equipotential  surfaces 
are  spherical  surfaces  with  centre  at  the  baric  centre.     Show 
that  the  whole  external  field  must  under  these  circumstances 
be  the  same  as  that  due  to  a  mass  equal  to  that  of  the  body, 
concentrated  at  0.     Show  that  if  at  internal  points  also  the 
line  of  action  of  the  force  always  passes  through  0,  the  density 
of  the  body  is  a  function  only  of  the  distance  from  0.     The 
centre  of  gravity  of  a  finite  centrobaric  distribution  is  the 
baric  centre.      A  distribution  cannot  be  centrobaric  unless 
every  axis  drawn  through  its  centre  of  gravity  is  a  principal 
axis.     If  for  any  finite  space  outside  it  a  body  is  centrobaric, 
it  must  be  centrobaric  for  all  the  rest  of  outside  space.     A 
distribution  which  consists  of  a  spherical  distribution  and  a 
distribution  the  potential  function  due  to  which  at  all  outside 
points  is  zero  is  evidently  centrobaric. 

87.  Show  that  if  0  is  a  fixed  origin  within  or  near  a 
distribution  M'  of  attracting  or  repelling  matter,  if  P'  is  any 
point  of  M'  a(J/,l  P  any  point  without  M  '  more  distant  from 
O  than  any  pcint  of  M'  is,  and  if  P  =  (x,  y,  z),  P'  =  (x1,  y'}  z'), 
OP  =  r,  OP'  ==  rf,  f  OP'  =  <j>  ;  the  value  at  P  of  the  potential 
function  due  tc>  M'  i«  equal  to 


M' 

- 


MISCELLANEOUS    PROBLEMS.  363 

Show  that  if  A,  B,  C,  and  /  are  the  moments  of  inertia  of  M' 
about  the  coordinate  axes  and  about  OP  respectively, 

A  +  B+C=   CCC2r'*dm'  and   /=  f  C  C  r'2  •  sin2<£.  dm', 

and  that  if  0  is  the  centre  of  gravity  of  M  ',  the  second  term 
of  the  development  vanishes  so  that 


If  M'  is  centrobaric  and  if  0  is  the  baric  centre,  V  is  a  func 
tion  of  r  only  and  the  coefficients  of  r  in  the  general  develop 
ment  are  to  be  considered  as  constants. 

88.  If  the  law  of  attraction  is  expressed  by  any  function, 
<£'(?•),  of  the  distance,  the  intensity  of  the  attraction  of  any 
homogeneous   solid,  estimated  in  a  given  direction,  at  any 

point  P,  is  expressed  by  the  surface  integral  (  <J>(r)  •  cos  A  •  dS, 

where  r  is  the  distance  from  P  of  any  point  on  the  surface 
bounding  the  solid,  dS  the  element  of  this  surface,  and  X  the 
angle  made  by  the  normal  to  the  element  with  the  given 
direction.  [Minchin.] 

89.  The  function  f  (xpy)   can  satisfy  Laplace's  Equation 
only  if  p  —  1,  or  —  1,  or  0. 

90.  The  invariable  line  which  joins  the  centres  (A0,  BQ)  of  two 
homogeneous  spheres,  A  and  B,  moving  under  their  mutual 
attraction,  revolves  with  uniform  angular  velocity,  w,  about 
the  centre  of  gravity,  C,  of  the  two.     One  of  the  spheres,  A, 
does  not  rotate,  but  every  line  in  it  remains  parallel  to  itself 
during  the  revolution.     Show  that  every  particle  of  A  moves 
in  a  circle  of  radius  equal  to  the  distance  of  A's  centre  from 
(7,  and  is  at  every  instant  at  the  end  of  a  diameter  parallel 
to  B0A0.     Under  these  circumstances  a  loose  particle  at  D  on 
A's  surface  must  in  general  be  constrained  to  keep  it  moving 
in  its  path. 


364  MISCELLANEOUS    PROBLEMS. 

If  we  denote  the  radius  of  A  by  a,  the  distances  £QA0)  CA0 
by  d  and  r,  and  the  mass  of  B  by  M,  the  resultant  force  on  a 
particle  of  mass  m  resting  on  A  at  D  [Fig.  123]  has  the 
intensity  mw2r  =  kmM/d2  and  a  direction  DT  parallel  to 
A0B0,  while  the  attraction  of  B  upon  the  particle  has  the  inten 
sity  kMmf  BQD  and  the  direction  DB$.  Show  that  if  a  is 
fairly  small  compared  with  d,  a  constraining  force  equal  to 
3  akMm  (sin  2  0)  /(2  o?3),  where  0  =  CAD,  must  be  exerted  on 
m  in  a  direction  perpendicular  to  ^10£>  to  prevent  its  sliding 
on  A's  surface. 

Assuming  A  to  be  the  earth,  of  mass  M'  and  radius  a,  and 
B,  the  moon,  of  mass  ¥TT  M ',  with  centre  distant  60  a  from 


FIG.  123. 

the  earth's  centre,  prove  that  the  maximum  horizontal  lunar 
tide-generating  force  on  the  earth's  surface  is  to  the  force 
of  terrestrial  gravitation  as  1  to  11,500,000,  nearly.  Find 
approximately  the  "  vertical  tide-generating  force "  at  the 
points  on  the  earth's  surface  nearest  and  farthest  from  the 
moon. 

[The  student  is  strongly  advised  to  read  in  this  connection 
Prof.  G.  H.  Darwin's  charming  Lowell  Lectures  on  the  Tides.] 

91.  Supposing  that  a  sphere  of  water  is  brought  together 
by  the  mutual  attractions  of  its  particles  from  a  state  of 
infinite  diffusion,  and  that  the  amount  of  work  done  by  these 
forces  is  sufficient  to  raise  the  temperature  of  the  sphere 


MISCELLANEOUS    PROBLEMS.  365 

1°  C.  Show  that  the  radius  of  the  sphere  is  about  one- 
fortieth  of  the  radius  of  the  earth,  if  the  earth's  radius  be 
637  x  106  centimetres,  and  if  one  water-gramme-centigrade- 
degree  be  equivalent  to  4.2  x  107  ergs.  [Minchin.] 

92.  The  value  at  any  point  (x,  y,  z)  of  the  potential  func 
tion  due  to  any  system  of  attracting  matter  at  a  finite  distance 
is  F,  the  forces  due  to  the  attraction  of  this  matter  at  any 
point  (x',  y\  z')  is  F',  the  value  at  this  point  of  the  potential 
function  F',  and  the  density  p.  Show  that 

V  -  F'*)dx'dij'dz' 


F°  =  ^-fff £ 

2^JJJ  nv- 


where  the  integration  takes  in  all  space. 

93.  Prove  that  the  rise  of  sea  level  in  a  shallow  sea  caused 
by  the  attraction  of  a  homogeneous  hemispherical  mountain 
of  radius  c  rising  from  it  with  its  base  at  sea  level,  is  approxi 
mately  p'c2 /2  pa,  where  p'  is  the  density  of  the  mass  of  the 
mountain,  p  the  mean  density  of  the  earth,  and  a  its  radius. 

94.  A  fixed  gravitating  sphere  is  partly  covered  by  an  ocean 
extending  over  the  northern  side  of  a  parallel  of  colatitude  A. 
A  distant  fixed  gravitating  body  M  is  situated  on  the  north 
axis  of  this  small  circle.     Prove  that  if  the  self-attraction  of 
the  ocean  be  neglected,  M  will  cause  a  rise  of  water  at  the 
north  pole  approximately  equal  to  *'sin2  JA,  where  *  is  what 
the  rise  would  be  if  the  whole  sphere  were  covered. 

95.  Show  that  if  a  finite  distribution  consists  of  m  units  of 
positive  matter  and  m  units  of  negative  matter,  anyhow  dis 
tributed,  it  is  possible  to  draw,  with  any  given  finite  point  as 
centre,  a  spherical  surface   so  large  that  the  whole  flow  of 
force  through  it,  reckoned  arithmetically,  shall  be  as  small  as 
we  please.     Prove  that  the  lines  of  force  are  all  closed. 

96.  Imagine  any  point  P  in  empty  space  near  a  distribution 
of  repelling  matter  to  be  taken  as  origin  of  a  system  of  orthog 
onal  Cartesian  coordinates  with  axis  of  z  coincident  with  the 


366  MISCELLANEOUS   PROBLEMS. 

normal  to  the  equipotential  surface  which,  passes  through  P. 
Fwill  then  be  given  by  an  equation  of  the  form  V—f(x,  y,  z), 
where  Dxf,  Dyf  vanish  at  P,  and  —  Dzf  is  the  force  F  in  the 
direction  of  the  z  axis.  If  Q  is  a  point  near  P  on  the  sec 
tion  of  the  surface  V  =  VP  made  by  the  xz  plane,  and  if  we 
denote  the  coordinates  of  Q  by  (Ax,  0,  A«),  the  radius  of 

~ 


curvature  at  P  of  this  section  is  A  *™n    7TT~  )»  an(^  A  z  is  in 

Ax  =  o 


general  of  higher  order  than  Ace. 


+  ^  Ax2  -  Dx*  V+  terms  of  higher  order. 

Since  VQ  =  FP,  and  Dx  V  vanishes  at  P,  Dx2  V  =  ~    Prove 

F  1 

similarly  that  Dy2  V  =  —  and  then,  by  Laplace's  Equation,  that 

n2F=  _*YI__A 

"^ 2.      *  I        T~>  I         7~> 


Illustrate  these  results  by  an  example. 

97.  If  a  distribution  of  active  matter  is  symmetrical  about 
a  straight  line  (the  axis  of  x)  and  if  r  represents  the  distance 
of  any  point  from  this  axis,  the  potential  function  involves 
r  and  x  only  and  the  equipotential  surfaces  are  surfaces  of 
revolution.  Consider  one  of  these  surfaces,  $0,  on  which  V 
has  the  value  F0,  and  let  the  "  flux  of  force  "  through  so  much 
of  $0  as  lies  between  some  fixed  plane  (x  =  je0)  perpendicular 
to  the  x  axis,  and  the  plane  x  =  x,  be  represented  by  the 
function  27r/x,  then  if  ds  is  the  element  of  the  generating 
curve  of  S0  between  x  and  x  +  Ax,  and  if  r  is  the  distance  of 
ds  from  the  x  axis,  the  area  of  the  strip  of  S0  between  x  and 
x  -f-  Ax  is  approximately  2  TT?-.  ds,  the  flux  of  force  through  it 
is  —  2  irr .  Dn  V-  ds,  and  this  flux  is  the  change  made  in  2  TT/X  by 
increasing  x  by  Ax.  We  may  write,  therefore,  -Dg/x,  =  —  r  -  Dn  F, 


MISCELLANEOUS    PROBLEMS.  367 

and,  if  a  is  the  angle  which  the  exterior  normal  to  ds  makes 
with  the  x  axis, 

Dsfj.  =  Z^  •  sin  a  —  Z>r/x  •  cos  a,  Dn  V=  Dx  V-  cos  a  +  Dr  V-  sin  a, 
and  the  equation  becomes 

sin  a  (D^  -  r-DrV)  -  cos  a  (Z>r/n  +  r-DxV)  =  0. 

If  this  equation  is  to  hold  everywhere  on  every  equipotential 
surface,  the  coefficients  of  sin  a  and  cos  a  must  vanish  and  /x 
is  determined  (apart  from  an  additive  constant  to  be  chosen  at 
pleasure)  by  the  equations  Dx/u  =  r  •  Dr  V,  Dr^  =  —  r  •  Dx  V. 

Show  that  the  values  of  /x  corresponding  to  the  three 
familiar  potential  functions  -  X&,  Mx/(t*  +  x2)*,  M/(r>  +  x2)* 
are  %XQ  ?•*,  Mr1/^  +  x2)*,  and  -  Mx/(i*  +  x2)-.  Discuss  the 
physical  meanings  of  these  results. 

The  function  ^  defined  above  is  sometimes  called  "  Stokes's 
Flux  Function."  It  is  clear  that  the  level  surfaces  of  the 
functions  V  and  /n,  both  of  which  are  symmetrical  about  the 
x  axis,  cut  each  other  orthogonally  and  that  the  generating  line 
of  any  level  surface  of  /n  is  a  line  of  force.  Although  any  func 
tion  of  /a  equated  to  a  constant  would  serve  to  represent  the 
forms  of  analytic  lines  of  force,  a  special  advantage  arises  from 
the  use  of  ^  itself  from  the  fact  that  if  ^  and  /A2  are  flux 
functions  corresponding  to  two  different  potential  functions, 
V-i  and  VZJ  due  to  two  distributions  of  matter,  Ml  and  1T2, 
symmetrical  about  the  x  axis,  ^  +  /u,2  is  a  flux  function  of 
Vi  +  Fo,  the  potential  function  due  to  M±  and  M2  existing 
together.  If  generating  lines  of  the  ^  surfaces  be  drawn 
in  a  plane,  for  the  numerical  values  «,  a  +  8,  a  +  2  8, 
a  +  3  8,  a  +  4  8,  etc.,  and  the  lines  of  the  ^  surfaces  for  the 
values  b,  b  +  8,  6  +  28,  6  +  38,  6  +  48,  etc.,  8  being  any  con 
venient  interval,  the  intersections  of  the  curves  ^  =  a  +  n&, 
yu,2  =  6  +  (m  —  n)  8  will  be  points  on  the  generating  lines  of 
the  surface  ^  +  /x2  =  a  +  b  +  raS.  If,  then,  we  fix  m  for  a 
moment  and  give  to  n  in  succession  different  integral  values, 


368  MISCELLANEOUS    PROBLEMS. 

we  may  get  points  enough  to  enable  us  to  draw  the  line 
Pi  +  /n2  =  a  -f-  b  -f-  wS  with  sufficient  accuracy.  This  graphi 
cal  method  of  drawing  lines  of  force  (or  equipotential  surfaces) 
has  proved  in  the  hands  of  Maxwell  and  others  extremely 
fruitful.  Draw  accurately  several  of  the  lines  of  force  due  to 
a  charge  20  and  a  charge  —  10  concentrated  at  points  4  inches 
apart. 

98.  (a)  Show  that  if  P,  P'  are  any  definite  pair  of  inverse 
points  distant  respectively  r  and  r'  from  the  centre  0  of  a 
spherical  surface  S  of  radius  a,  the  ratio  PQ /P'Q  is  equal  to 
the  constant  a/r'  wherever  on  S  the. point  Q  may  be.    Hence 
show  that  if  V  is  the  potential  function  due  to  a  heterogeneous 
surface  distribution  on  S, 

VP,  =  VP(a/r')zndDr,(VP)  =  -  as-DrVp./r'3  -  aV/r<*. 

(b)  Prove  that  i*'*>Drr2,  +  r'*'*.Dr,rj,=  -(aVr>  FP.)1/2. 

[Routh.] 

(c)  Prove  that  as  both  r  and  r'  are  made  to  approach  a, 
limit  (Dr  VP  +  Dr>  •  F»  =  -  V/a.     [Stokes.] 

99.  If  P,  P'  are  any  definite  pair  of  inverse  points  with 
respect  to  a  right  section  of  an  infinitely  long  cylindrical  sur 
face  of  revolution,  and  if   Q  be  any  variable  point  on  the 
circumference,  P'Q/PQ  is  equal  to  the  constant  r'/a.     Show 
that  if  the  cylinder  be  covered  with  a  superficial  distribution 
the  density  of  which  varies  from  filament  to  filament  of  the 
surface,  VP,  —  VP  =  2  log  (r  '/a)  •  M,  where  M  is  the  amount  of 
matter  on  the  unit  length  of  the  surface. 

100.  V  is  the  potential  function  due  to  a  volume  distribu 
tion  of  density  p  in  the  region  T  and  a  surface  distribution  of 
density  a-  on  the  surface  S.      V  is  the  potential  function  due 
to  a  volume  distribution  of  density  p'  in  the  region  T1  and  a 
surface  distribution  of  density  a-'  on  the*  surf  ace  S'.    Using  all 
space  as  the  field  of  Green's  Theorem,  apply  [145]  to  these 
functions  and  interpret  the  resulting  equation  as  giving  an 
expression  for  the  mutual  energy  of  the  distributions. 


MISCELLANEOUS    PROBLEMS.  369 

101.  If  two  systems  of  matter  (If  and  M'),  both  shut  in  by 
a  closed  surface  S,  give  rise  to  potential  functions  (  V  and  F'), 
which  have  equal  values  at  every  point  of  S,  whether  or  not 
S  is  an  equipotential  surface  of  either  system,  then  V  can 
not  differ  from  V  at  any  point  outside  S,  and  the  algebraic 
sum  of  the  matter  of  either  system  is  equal  to  that  of  the 
other. 

102.  Prove  that  the  level  lines  of  the  function  u  =  F^  (x,  y,  z) 
on  the  surface  FI  (a*,  y,  z)  =  0  have  direction  cosines  which  are 
to  each  other  as 


and  (Dxf\  .  DyF2  -  DyF,  .  DXF2)  • 

and  that  if  these  quantities  be  represented  by  A,  /*,  and  v, 
respectively,  the  direction  cosines,  at  the  point  (x,  y,  z),  of  a 
curve  which  lies  on  Fl  and  cuts  orthogonally  at  that  point  a 
level  line  of  u  on  the  surface,  are  to  each  other  as 

0  •  DzFi  -  v  •  DyF,)  :  (v  •  DXF,  -  A  •  DZF,}  :  (A  -  DyF,  -  p.  -  DXF,}. 


In  particular,  if  u  =  x/z  and  if  Fl(x,  y,  z)  =  x*(b2  —  ?/2)—  «2-2, 
the  level  lines  of  u  on  Fl  are  straight  lines,  the  direction 
cosines  of  which  at  any  point  P  are  in  the  ratio  up  :  0  :  1, 
and  since  the  sum  of  the  squares  of  these  cosines  must  be 
equal  tojunity,  the  cosines  themselves  are  u  /  V*r  +  1,  0,  and 


103.  In  the  case  of  a  columnar  distribution  the  density  of 
which  varies  only  with  the  distance  r  from  a  fixed  axis,  the 
lines  of  force  are  straight  lines  radiating  from  the  axis  (Sec 
tion  34),  and  the  potential  function  Fand  the  resultant  force 
Dr  V  are  functions  of  r  alone.  If  we  apply  Gauss's  Theorem 
to  a  cylindrical  surface  of  revolution  S,  coaxial  with  the  distribu 
tion,  we  learn  that  2 -n-r DrV  =  4 TT  times  the  mass  M  of  the 
unit  length  of  so  much  of  the  distribution  as  is  enclosed  by  S. 


370  MISCELLANEOUS   PROBLEMS. 

Show  that  if  the  distribution  is  a  solid  homogeneous  repell 
ing  cylinder  of  radius  a  and  density  p,  Dr  V  =  2  irpr  and 
V=TTp  [r2  —  a2  +  2  a2  log  a],  if  r  is  less  than  a.  If  r  is  greater 
than  a,  Dr  V  —  2  7rpa2/r  and  V  =  2  Trpa?  log  r.  Show  also  that 
if  the  distribution  is  merely  a  surface  charge  of  density  a-  on 
a  cylindrical  surface  of  radius  a,  V  =  4  Ti-ao-  log  a  within  the 
cylinder,  and  V  =  4  TTO-O-  log  r  without. 

104.  If  V  is  the  gravitational  potential  function  belonging 
to  a  given  distribution  M  of  attracting  matter,  and  if  k  is  the 
constant  of  gravitation,  the  force  of  gravitation  at  any  point 
in  any  direction  s,  measured  in  dynes,  is  the  value  at  that  point 
of  k  •  Ds  For  Ds  F0,  where  F0  =  k  V  ;  and  V2  F0  =  —  4  irkp.  Prove 
that  if  M  be  made  to  rotate  about  the  axis  of  z  with  constant 
angular  velocity  w,  if  O  =  £  w2(x2  +  y2),  and  if  F'  =  F0  +  O, 
the  apparent  force  at  any  point  in  any  direction  s  is  Ds  F'  and 
V2  F'  =  —  4  irkp  4-  2  or.  Prove  also  that  if  S  represents  the 
surface  of  M,  and  n  a  normal  to  the  surface  drawn  inwards, 
if  v  is  the  volume  of  the  distribution,  and  if  p0  is  its  mean 
density, 

-  4  Trkvp,  +  2  «fy  =  - 


[E.  S.  Woodward,  "  The  Gravitational  Constant  and  the  Mean 
Density  of  the  Earth/'  Astronomical  Journal,  1898.] 

If  M  represents  the  earth,  a  the  semiaxis  major,  and  e  the 
eccentricity  of  the  generating  ellipse  of  the  earth's  spheroid, 
<f>  and  \  the  latitude  and  longitude  of  dS,  we  have 


v  =  ±  ^a3  Vl  -  e2,  dS  =  a2  (1  -  e2)  cos 


and  the  acceleration  Dn  F'  is  given  in  centimetres  per  second  per 
second  by  the  equation  Dn  V  =  a-f  (3  sin2  <j>  +  y  cos2  <£  •  cos  2  A, 
where  a  =  978,  ft  =  5.19,  and  y  is  a  constant,  so  that 


MISCELLANEOUS    PROBLEMS.  371 

Show  that  if  x  =  e  sin  <£. 

2ff2(i-e2)  rc    df      r-17    _ 

e         Jo  (1  —  x-)-Jo  TF 

and 


and,  assuming  that  log  e-  =  3.83050,  log  a  =  8.80470,  obtain 
Professor  Woodward's  equation,  Ay>0  =  36797  x  10  ~n. 

For  a  discussion  of  the  value  of  p0,  see  Prof.  J.  H.  Poynting's 
Adams  Prize  Essay  on  "  The  Mean  Density  of  the  Earth." 

105.  If  u  is  single-valued  and  harmonic  at  all  points  of  a 
region  but  one  (the  exceptional  point  being  an  interior  point 
P),  and  if  u  becomes  infinite  for  all  paths  along  which  the 
point  (a;,  y)  approaches  P,  then  u  can  be  written  in  the  form 
u  —  a- log  r  -f-  r,  where  v  is  single- valued  and  harmonic  at  all 
points  of  the  region.     [Bocher.] 

106.  If  the  superficial  density  of  a  mass  distributed  on  a 
spherical  surface  is  inversely  proportional  to  the  cube  of  the 
distance  from  a  fixed  point  A,  the  distribution  is  centrobaric. 
If  A  is  inside  the  surface,  it  is  the  baric  centre  ;  if  A  is  outside, 
its  inverse  point  is  the  baric  centre. 

107.  If  the  superficial  density  of  a  columnar  distribution  on 
a  cylindrical  surface  of  revolution  varies   inversely  as  the 
square  of  the  distance  from  a  given  line  parallel  to  the  axis 
of  the  cylinder,  there  is  a  baric  line  within  the  distribution 
parallel  to  the  axis.     Where  is  this  line  ? 

108.  A  certain  distribution  M  has  two  mutually  exclusive 
closed  equipotential  surfaces,  Sl  and  S2,  upon  which  V  has  the 
different  constant  values  Ci  and  <72.     Will  the  potential  func 
tion  in  space  without  /S^  and  S2  be  the  same  for  the  original 
distribution  and  for  distributions  on   ^  and   S    of  surface 


372  MISCELLANEOUS    PROBLEMS. 

density  a-  =  —  DH  F/4  ?r  together  with  so  much  of  M  as  lies 
without  the  surfaces  ? 

109.  The  straight-line  tangents  at  a  point  to  a  tube  of  force 
which  ends  there,  evidently  form  a  cone  of  definite  solid  angle. 
A  number  of  points,  Ply  P2,  Ps,  etc.,  have  charges,  m1}  m2,  w3, 
etc.     Show  that  if  at  any  one  of  these  points  there  end  two 
tubes  the  solid  angles  of  the  cones  of  which  are  o>  and  CD',  the 
flow  of  force  in  the  one  tube  is  to  the  flow  of  force  in  the  other 
as  W  :  to'.     Show  also  that  if  a  tube  starts  with  solid  angle  wk 
at  a  point  Pk  where  the  charge  is  mk)  and  ends  with  solid  angle 
cur  at  a  point  Pr  where  the  charge  is  mr)  o>kmk  is  numerically 
equal  to  c>rmr. 

110.  All  the  masses  of  a  certain  distribution  lie  within  two 
closed  surfaces  S1  and  S2J  which  exclude  each  other  and  are 
equipotential.     All  the  lines  of  force  which  abut  on  a  con 
tinuous  portion  A  of  S1  also  abut  on  S2.      All  the  lines  of 
force  which  abut  on  S{  outside  of  A  are  open,  while  none  of 
the  lines  which  abut  on  S2  are  open.     Show  that  one  of  the 
equipotential  surfaces  is  made  up  of  two  lobes,  one  of  which 
includes  S2  alone  and  the  other  both  S1  and  S2.     Separating 
the   closed  lines  of  force  from  the  open  ones  is  a  surface 
which  passes  through  the  point  where  the  lobes  of  the  sur 
face   just  mentioned  are  connected.     All  the   equipotential 
surfaces  are  closed. 

111.  The  potential  function  due  to  a  certain  distribution  of 
matter  has  a  value  at  any  point  Q  which  depends  only  upon 
the  distance,  r,  of  Q  from  a  fixed  point  0.     This  value  is 


2  a*\ 

-r2-  —  J, 


3r 


or 


according  as  0  <  r  <  a,  a<r<bj  b<r<c,  or.r>c.     What  is 
the  distribution  ?     [p  —  0,  p  —  1,  p  —  0,  a  =  1,  p  =  0.] 


MISCELLANEOUS    PROBLEMS.  373 

112.  The  lines  of  force  due  to  two  similar,  homogeneous, 
infinitely  long,  straight  filaments  of  repelling  matter,  parallel 
to  the  z  axis  and  cutting  the  xy  plane  at  the  points  (a,  0), 
(—a,  0)  are  hyperbolas  of  the  family  x*  —  2  \xy  —  if  =  a2. 

113.  (a)  Prove  that  when  there  is  symmetry  about  the  axis 

from  which  0  is  measured,  >•»'  .  Pm  (cos  0)  and  Pm          ^  ,  where 


Pm(cos  0)  is  the  coefficient  of  am  in  the  development,  in  ascend 

ing  powers  of  a,  of  (1  -  2  a  cos  6  +  a2)"*,  are  particular  solu 
tions  of  Laplace's  Equation  in  polar  coordinates  ;  that  is,  of 

r  -  Dr\r  S>.+gjj-  A(sin  0  •  De  V)  =  0. 

Hence  show  that  any  expression  of  the  form 
A0  +  A,r  •  P^cos  0)  +  A.2>*  •  P2  (cos  0)  +  •  •  -  4-  Aj»  •  Pn(cos  0) 

,  BQ  ,  JVP^cosfl)   ,  ^8.P,(cosg)  £m-Pm(cos6) 

'    /•  ^  ~7~  r»-M 

where  J0,  ^0,  ^1;  J51?  J2,  P2,  etc.,  are  arbitrary  constants, 
satisfies  the  equation.  The  P's  here  introduced  are  some 
times  called  Legendre's  Coefficients,  sometimes  Zonal  Surface 
Spherical  Harmonics. 

(b)  Show  that  P0(/x)  =  1, 


(c)  Show  that  when  (9-0,  Pm(cos  tf)  or  Pm(l),  the  coefficient 
of  am  in  the  development  of  (1  —  a)  -1,  is  equal  to  1. 

114.  Prove  that  if  in  any  case  of  symmetry  about  a  line, 
a  convergent  series  a0  +  a^  +  a2z2  +a3z3  -\  ----  represents  the 


374  MISCELLANEOUS    PROBLEMS. 

value  of  the  potential  function  at  a  point  Q  distant  z  from  the 
fixed  point  0,  both  0  and  Q  being  on  the  line  ;  then  the  series 

a0P0(cos  6)  +  a^P^cos  0)  +  a2r*P2(cos  0)  -f  asi*Ps(co8  6)  H , 

formed  by  writing  instead  of  zm  in  the  former  series  r™  •  Pm(cos  6), 
will  represent  in  polar  coordinates  with  0  as  origin  and  the 
given  line  as  axis  from  which  6  is  measured,  a  finite,  single- 
valued  function  which  satisfies  Laplace's  Equation  and  for  all 
points  on  the  given  line  on  the  positive  side  of  0,  where  6  =  0 
and  r  =  z,  has  the  same  values  as  the  given  series.  Given  the 
radius  of  convergence  of  the  first  series,  within  what  limits 
can  we  safely  use  the  second  series  ?  If  any  portion  of  the 
given  line  traverses  a  region  of  empty  space,  does  the  new 
series  represent  the  potential  function  at  all  points  in  this 
region  within  the  limits  of  convergency  of  the  series  ? 

115.    Prove  that  if  in  any  case  of  symmetry  about  a  line,  a 

convergent  series  —  +  -f  +  -f  -f  •  •  •  represents  the  value  of  the 
potential  function,  the  series 


0)  ,  ftgP^cos  6)      a3P2(cos  0) 


formed  by  writing  instead  of  -^  in  the  former  series,    "' 


will  represent,  so  long  as  the  new  series  is  convergent,  a  finite, 
single-valued  function  which  satisfies  Laplace's  Equation  and, 
for  all  points  on  the  line  on  the  positive  side  of  0,  has  the 
same  values  as  the  given  series. 

116.  Prove  that  the  potential  function  due  to  a  uniform  circular 
ring  of  mass  M,  of  radius  a,  and  of  small  cross-section,  is  equal  to 


M(    _1   ft2.P2(cos(9)      1-3   a4.P4(cos(9)  _       \ 
r\        2"         r2  2-  4"         r4  "/ 

if  aO,  and  equal  to 


r'-P^cosfl)      1-3  r4-P4(cosfl) 
a"        ~  +  2-4'~ 


COS0)  \ 

') 


MISCELLANEOUS    PROBLEMS.  375 

if  a  >  r,  where  the  centre  of  the  ring  is  the  origin,  and  the  axis 
of  the  ring  the  axis  from  which  0  is  measured. 

117.  Prove  that  the  potential  function  due  to  a  uniform 
circular  disc  of  mass  Jf,  of  radius  a,  and  of  small  thickness,  is 
equal  to 

2Jfcf/l   a2         1       a4.Pg(cosfl)        1-3     ft6 .  P4  (cos  6) 
a*   \2'  r      22-2!"         r3  "f~23.3!"          i* 

if  a  <  ?-,  and  to 


2 

if  a  >  r,  when  the  centre  of  the  ring  is  the  origin. 

118.  Show  that  the  expression  ±(r2  —  c2  -f-  y*)  /y  of  equa 
tion  [21],  page  12,  is  numerically  equal  to  the  length,  k,  of 
the  chord  of  the  sphere,  formed  by  a  radius  vector  drawn  from 
P  to  a  point  L  on  the  surface,  distant  y  from  P.     The  sign  is 
to  be  taken  negative  or  positive,  according  as  L  is  or  is  not 
visible  from  P.     Hence  find  an  expression,  Trcra(kl  ±  AVj/c2, 
for  the  intensity  of  the  attraction  of  an  "  annulus  "  of  a  thin 
spherical  shell  lying  between  two  parallels  of  latitude,  at  any 
point  P  on  the  axis. 

119.  A  thin  spherical  shell  of  radius  a  attracts  an  internal 
particle  P  at  a  distance  c  from  its  centre.     If  the  shell  be 
divided  into  two  parts  by  a  plane  through  P  perpendicular 
to  the  radius,  the  resultant  attraction  of  each  part  at  P  is 
2  Trcra  \_a  —  Va2  —  c2]  /c2.     [Todhunter's  History  of  Attraction.] 

120.  The  equation  of  the  surface  of  an  infinitely  long  homo 
geneous  cylinder  of  density  p,  the  lines  of  which  are  parallel 
to  the  z  axis,  being  r=/(0),  a  filament  of  the  cylinder  of 
cross-section  rdrdO  contributes  to  the  components  (X,  T)  of 
the  attraction  at  the  origin  the  amounts  2  p  cos  0  •  dr  •  dO  and 
2  p  sin  0  •  dr  -  dO   respectively.       If    the    cross-section   of  the 
cylinder  is  an  ellipse  of  semiaxes  a  and  b  and  if  the  origin 
is  on  the  surface  and  distant  y0,  x0  respectively  from  the 


376  MISCELLANEOUS    PROBLEMS. 

principal  planes,  the  equation  of  the  surface  may  be  written 
in  the  form 

r  =  2  (b2x0  cos  6  +  azy0  sin  0)  /  (b2  cos2  0  +  a2  sin2 0). 
Assuming  that 

C        f    .     =^r.-X/*.tan 
Ja  +  btSitfx       a  —  b  |_          A  c*- 

prove  that  in  this  case 

X  =  4  Trpbx0/(a  +  5),    F  =  4  7rpay0  /  (a  +  b) 

and  that  the  resultant  force  has  the  intensity  4  Tlf/  (a  +  b), 
where  M  is  the  mass  of  the  unit  length  of  the  cylinder. 
Prove  also,  by  a  method  analogous  to  that  of  Section  12, 
that  the  attraction  due  to  a  homogeneous  shell  bounded  by 
two  concentric,  similar,  and  similarly  placed  elliptic  cylin 
drical  surfaces  is  zero  within  the  shell,  and  that  the  attraction 
components  (X,  Y)  at  any  point  within  a  solid  homogeneous 
elliptic  cylinder  are  proportional  to  x  and  y  respectively. 

121.  If  two  confocal  ellipses  (s  and  s')  have  semiaxes  (a,  b) 
and  (a1,  b')t  a  point  (cc,  y)  on  s  is  said  to  correspond  to  a  point 
(x'j  y)  on  s',  if  x/x'  =  a /a'  and  y/y'  =  #/&'.  Show  that  if 
P!  and  P2  are  any  two  points  on  s  and  P/  and  P2',  the  corre 
sponding  points  on  s',  P-^PJ  =  P2Pi'.  Hence  prove  (Section  51) 
that,  if  two  homogeneous,  solid,  confocal,  elliptic  cylinders  of 
the  same  density  be  divided  into  corresponding  thin  strips  by 
planes  parallel  to  the  xz  plane,  the  x  component  of  the  attrac 
tion  of  any  strip  of  the  first  at  a  point  P'  on  the  second,  is  to 
the  x  component  of  the  attraction  of  the  corresponding  strip 
of  the  second  at  a  point  P  on  the  first  corresponding  to  Pf, 
as  b  is  to  b'.  The  two  components  of  the  attraction  of  the 
whole  of  the  first  cylinder  at  P'  are  to  the  same  components 
of  the  attraction  of  the  second  cylinder  at  P,  as  b  to  b'  and  as 
a  to  a'. 


MISCELLANEOUS    PROBLEMS. 


377 


122.  It  follows  from  the  results  stated  in  the  last  two 
problems,  that  if  the  components  at  an  outside  point  Q'  of 
the  attraction  due  to  a  solid  homogeneous  elliptic  cylinder  of 
density  p  bounded  by  the  surface  s  (Fig.  124)  be  X  and  F,  if 
a  surface  s'  confocal  to  s  be  drawn  through  Q',  and  if  X  and  Y 
are  the  components,  at  Q  on  s  which  corresponds  to  Q'  on  s't 
of  the  attraction  of  a  cylinder  of  density  p  bounded  by  s' ; 
X/X=b/b\  Y/Y=a/a',  where  a  and  b  are  the  semi- 
axes  of  s,  and  a'  and  b'  those  of  s'.  Show  that  A",  Y  are  the 
components  at  Q  of  the  attraction  due  to  a  cylinder  of 


FIG.  124. 

density  p,  bounded  by  a  surface  s"  drawn  through  Q,  similar 
to  s'.  Show  also  that,  if  the  coordinates  of  Q  are  x,  i/, 

X'  =  ±7rpbx/(a'  +  b'),     Y'  =  4  irpay/(a'  +  £'). 

Prove  that,  if  a  =  o,  b  =  3,  x'  =  4,  and  ij  =  if- ;  x  =  ™, 
y  =  V5,  «.'  =  6,  b'  =  V20,  so  that,  approximately,  A"'  =  12p, 
Y  =  13  -  42  •  p. 

123.  Two  parallel  planes,  the  direction  cosines  of  the  nor 
mals  to  which  are  (7,  m,  n),  touch  two  confocal  ellipsoidal 
surfaces  at  the  points  PI,  PI  respectively.  The  serniaxes  of 


378  MISCELLANEOUS    PROBLEMS. 

the  surfaces  are  (a,  b,  c)  and  (a  -\-  da,  b  +  db,  c  +  do)  where, 
since  they  are  confocal,  d(a2)  =  d(b2)  =  d(c2).  Show  that  if 
p  and  p  -f-  dp  are  the  lengths  of  the  perpendiculars  dropped 
from  the  origin  on  the  tangent  planes,  p2  =  a2l2  +  b2m2  -f  c2n2, 
and p.dp  =  l2 .d(a2)  +  m?  -d(b2)  +  n2 •  d(c2)  =  d(a2),  so  that  dp 
is  inversely  proportional  to  p.  If  the  surfaces  bound  a  homo 
geneous  shell,  this  is  called  a  thin  focaloid.  Show  that  the 
thickness  of  the  shell  at  the  point  P  differs  from  dp,  if  at 
all,  by  an  infinitesimal  of  higher  order,  and  that  a  superficial 
distribution  on  an  ellipsoid  with  surface  density  inversely 
proportional  to  p  is  equivalent  to  a  thin  focaloid  bounded 
internally  by  the  surface.  The  thickness  of  a  thin  homoeoid 
at  any  point  is  directly  proportional  to  p. 

124.  Show  that  if  the  potential  function  due  to  a  distribution 
of  matter  has  the  value  zero  at  all  points  outside  the  ellipsoid 
Lx2  +  My2  +  Nz2  =  1  and  the  value  /*  (1  -  Lx2  -  Mx2  -  Nz2) 
at  all  inside  points,  the  distribution  consists  of  a  homogeneous 
ellipsoid   of   density  /x,  (L  -f  M  +  N)  /  2  TT  and   a   superficial 
stratum  on  it  of  surface  density  —  n/Zirp,  where  p  is  the 
length  of  the  perpendicular  dropped  from  the  origin  on  the 
tangent  plane.     Since  the  surface  distribution  is  equivalent 
to  a  thin  focaloid,  it  is  clear  that  the  potential  function  due 
to  a  homogeneous  ellipsoid  has  at  outside  points  the  same 
values  as  the  potential  function  due  to  a  thin  focaloid  of  the 
same  mass  coincident  with  the  surface  of  the  ellipsoid.    Prove 
from  this  that  confocal  ellipsoids  of  equal  mass  have  equal 
potential  functions  at  points  outside  both. 

125.  Two  homogeneous,  solid,  confocal  ellipsoids  of  masses 
MI  and  M2  attract  any  particle  outside  both  with  forces  which 
have  the  same  direction  and  are  to  each  other  as  J/i  to  Jf2. 
[Maclaurin.] 

126.  Show  that  it  follows  from  the  reasoning  on  pages  123 
and  124  that  the  components,  taken  parallel  to  the  axes,  of 
the  attraction  of  a  homogeneous  ellipsoid  S  at  any  point  P' 
on  the  surface  of  a  homogeneous  confocal  ellipsoid  S1  of  the 


MISCELLANEOUS    PROBLEMS.  379 

same  density,  are  to  the  corresponding  attraction  components 
due  to  S'  at  the  point  P  on  S  which  corresponds  to  P',  as 
the  areas  of  the  principal  sections  of  S  and  S'  perpendicular 
to  these  components.  [Ivory.] 

127.  We  know  from  the  equations  of  page  191  that,  in  the 
case  of  a  prolate  ellipsoid  uniformly  polarized  in  the  direction 
of  the  long  axis,  the  depolarizing  force  is 


Prove  that  if  the  ratio  of  a  to  b  is  large,  this  is  nearly  equal  to 
-  4  7rA(b2/a2)  [log (2  a/b)  -  1],  and  that  when  a/b  =  4,  this 
approximate  result  is  in  error  by  about  4  per  cent. 

Show  that  if  we  denote  the  depolarizing  force  in  an  ellipsoid 
of  revolution  uniformly  polarized  in  the  direction  of  the  x  axis 
by  \A,  A  has  the  values  12.57,  6.63,  5.16,  4.19,  2.18,  0.95,  0.25, 
0.0054,  0.0016,  0.0004,  according  as  a/b  is  equal  to  0,  J,  j,  1, 
2,  4,  10,  100,  200,  400. 

128.  If  the  quantity  c  on  page  121  be  supposed  to  increase 
without  limit,  the  limits  of  the  expressions  for  X  and  Y  are 
the  force  components  within  a  homogeneous  elliptic  cylinder 
of  semiaxes  a  and  b.  Making  use  of  the  integral 

dx  2  (x  +  #)* 


show  that  these  limits  are  4  irbpx  /  (a  +  b)  and  4  irapy  /  (a  +  b). 
Using  the  form  of  integral  given  on  page  124  in  the  seventh 
line  from  the  bottom,  show  that  if  c  be  made  to  increase 
without  limit,  the  limit  of  X  is  —  4*pftz/(af  -f-  b1)  and  that 
the  corresponding  limit  of  Fis  —  ±irpay /(a'  -f  b'). 

Show  that  the  equipotential  surfaces  within  an  infinitely 
long,  solid,  homogeneous,  elliptic  cylinder,  the  semiaxes  of 
which  are  a  and  b,  are  elliptic  cylindrical  surfaces,  the  ratio 
of  the  semiaxes  of  any  one  of  which  is  va/  vS 


380  MISCELLANEOUS    PROBLEMS. 

129,  Using  the  integrals  given  on  page  190,  show  that  if 
a  =  b  >  c  and  if  X2  =  (a2  —  c2)/c2,  we  may  write  the  expres 
sions   for  the  attraction  components  within  a  homogeneous 
oblate  ellipsoid  of  revolution,  in  the  form 

(-  3  Jfz/2  A'V)  [tan-1  A  -  A/ (14-  A2)], 
(-  3  My/ 2  A-V)  [tan"1  A  -  A/(l  +  A2)], 
(-  3  Mz/\3cs)(\  -  tan-1  A). 

130.  Show  that  if  in  the  case  of  the  prolate  ellipsoid  of 
revolution  where  b  =  c  <  a,  we  put  A  =  ea/c,  the  components 
of  attraction  at  the  inside  point  (x,  ?/,  z)  may  be  written 


(3  Jfaj/AV)  [A/  Vl  +  A2  -  log  (A  4-  Vl  4-  A2)], 
(3  My  12  AV)  [log  (A  +  Vl  +  A2)  -  A  Vl  +  A2], 
(3Mz/2  A3c3)  [log  (A  4-  Vl  +  A2)  -  A  VT+72]. 

131.  If  these  force  components  be  denoted  by  X,  Y}  Z,  the 
quantity  (X  j  x  4-  Y  /y  4-  Z  /z)  is  numerically  equal  to  —  4  ?r/o 
within  any  ellipsoid  of  revolution.     This  is  true  in  the  case 
of  every  ellipsoid,  as  Poisson's  Equation  shows. 

132.  If  a  =  a2,  (3  =  b2,  y  =  c2,  and  if  G0  has  the  value  given 
on  page  122, 

a-DaG0  4  p-DfiGt  +  y  •  DyG,  =  -  t  G() 

and  2  (a  -  ft)  DaD8  G0  =  Da  G0  -  D&  G{, 

The  potential  function  V  satisfies  the  equation 


at  every  point  within  a  homogeneous  ellipsoid. 

If  (a',  b',  c')  are  the  semiaxes  of  an  ellipsoidal  surface  through 
a  point  (x,  y,  ,?),  confocal  with  the  ellipsoid  E  which  has  the 
semiaxes  (a,  b,  c),  and  if  the  result  of  substituting  a',  b',  c'  for 
a,  b,  c  in  the  expression  for  G0  be  denoted  by  G',  the  value  of 


MISCELLANEOUS    PROBLEMS.  381 

the  potential  function  due  to  E  at  an  external  point  may  be 

written 

f  J/  \  G'  +  2  x2  •  DtG'  +  2y*.DmG'  +  2z2.  DnG'  \ 

where  I  =  a'-,  m  =  b*2,  n  ==  c".  [Tarleton.] 

133.  Show  that  if  X,  Y,  Z  are  the  components  of  the  body 
forces  applied  to  a  mass  M  of  liquid  revolving  with  uniform 
angular  velocity  <u  about  the  axis  of  z,  and  if  p  denotes  the 
pressure  at  the  point  (x,  y,  z), 

dp=(X  +  o>-x)dx  +(T+  <*ry)dy  +  Z  dz, 
so  that  at  a  free  surface 

(X  +  «?x)dx  +(Y+  «rt/)dy  +  Zdz  =  0. 

134.  If   the   liquid   be    homogeneous    and  exposed  to  its 
own  attraction  only,  and  if  the  bounding  surface  be  the  ellip 
soid  tf<?y?  +  a-(?if  4-  a-b2z2  =  a-lrc2,    we   have   X=- 

Y  =  —  |  J/Z>0?/,  Z—  —  \  J/J/o^,  and  at  the  free  surface 

Ircrxdx  +  a-rydy  -h  arlrzdz  =  0, 
so  that 


Show  that  this  condition  is  satisfied  for  a  given  value  of  to  by 
an  oblate  ellipsoid  of  revolution  (Example  129)  for  which  X 
satisfies  the  equation,  X  =  tan  [(3  X  +  2  a>2X3/4  irp)  /(3  +  X2)]  ; 
but  that  a  prolate  ellipsoid  of  revolution  is  not  a  possible 
form  of  the  bounding  surface.  [Besant's  Hydromechanics, 
Vol.  I;  Laplace's  Mecanique  Celeste,  Vol.  III.] 

135.  Prove  that  if  V  be  the  potential  function  due  to  any 
distribution  of  matter  over  a  closed  surface  $,  and  if  a-'  be 
the  density  of  a  superficial  distribution  on  S,  which  gives  rise 
to  the  same  value,  F',  of  the  potential  function  at  each  point 
of  S  as  that  of  a  unit  of  matter  concentrated  at  any  given 
point  0,  then  the  value  at  0  of  the  potential  function  due  to 

the  first  distribution  is    j  V1  -  a-  •  dS. 


382  MISCELLANEOUS  PROBLEMS. 

1/136.  Show  that  the  derivative  of  the  function  x2  +  xy  +  z* 
at  the  point  (1,  2,  3)  in  the  direction  denned  by  the  cosines 
(j,  i,  $  V2)  is  J(5  +  6  V2).  Find  the  angle  between  the 
vector  differential  parameter  of  this  function  and  the  direction 
just  defined,  at  any  point  of  the  plane  3  x  -+-  y  -f-  2  »  A/2  =  0, 
at  every  point  of  the  line  x  +  y  =  0,  x  =  2z,  and  at  the  origin. 
Show  that  it  is  not  possible  to  find  a  scalar  function  the  level 
surfaces  of  which  cut  orthogonally  the  lines  of  the  vector 
(x  +  y,  z,  y).  Show  that  the  normal  derivative  of  the  function 
x*  +  y  +  %  with  respect  to  the  function  x  +  y  -f-  2  is  zero  at 
every  point  of  the  plane  x  =  —  1.  Prove  that  if  u  and  v  are 
the  distances  of  the  point  (x,  y,  z)  from  two  fixed  points, 
hu  =  A,  -  1. 

»,  137.  A  harmonic  function  which  has  a  constant  gradient 
different  from  zero  cannot  vanish  at  infinity  like  the  Newtonian 
Potential  Function  due  to  a  finite  mass. 

1  138.  [f(x,  y,  s),  0,  0],  [3>(x),  0,  0],  [*(y,-«),  0,  0],  the  first 
of  which  is  neither  lamellar  nor  solenoidal,  the  second  lamel 
lar  but  not  solenoidal,  and  the  third  solenoidal  but  not  lamellar, 
are  examples  of  vectors  the  lines  of  which  are  parallel  straight 
lines,  though  the  intensities  are  not  constant.  Prove  that  if 
in  any  region  the  lines  of  a  vector  which  is  both  lamellar  and 
solenoidal  are  parallel  straight  lines,  the  intensity  of  the  vector 
is  everywhere  in  that  region  the  same. 

*  139.  (2x/r,  2  y/r,  2z/r)  and  (sin  y,  A/3,  cos  y),  the  first  of 
which  is  lamellar  but  not  solenoidal  and  the  second  solenoidal 
but  not  lamellar,  are  examples  of  vectors  with  constant  inten 
sities,  which  have  lines  which  are  not  straight  lines  parallel 
to  each  other.  Prove  that  if  the  lines  of  a  lamellar  point 
function  which  has  a  constant  tensor  are  parallel  straight 
lines,  the  vector  is  solenoidal.  Prove  also  that  if  the  lines  of 
a  solenoidal  vector  point  function  which  has  a  constant  tensor 
are  parallel  straight  lines,  the  vector  is  lamellar. 
'  140.  The  vectors  (x  +  2  zy,  y  +  3  xz,  xy),  (2  zy,  3  xz,  xy  +  2z) 
have  everywhere  equal  divergences  and  curls  and  their  tensors 


MISCELLANEOUS    PROBLEMS.  383 

are  equal  all  over  the  surface  x2  +  y-  —  4  z-  +  6  xyz  —  0.  It 
is  evident,  therefore,  that  such  vectors  as  these  are  not  deter 
mined  when  their  curls  and  divergences  are  given.  What 
additional  information  would  determine  an  analytic  vector 
which  does  not  vanish  at  infinity  ?  The  scalar  potential 
function  of  a  certain  vector  has  the  value  unity  from  r  =  0  to 
r  —  1,  where  r2  =  or  +  if  +  z-  •  and  the  value  1  /r  from  r  =  1 
to  r  =  oo .  Is  the  vector  everywhere  solenoidal  and  lamellar  ? 
Can  you  determine  an  everywhere  lamellar  and  solenoidal 
vector  which  has  the  value  13  at  infinity? 
/^l41.  If  at  any  surface  the  normal  component  or  a  tangential 
component  of  a  vector  is  discontinuous,  must  we  suppose  that 
there  is  divergence  at  the  surface  ?  Illustrate  your  answer  by 
a  simple  numerical  illustration. 

142.  S  is  a  portion  of  an  analytic  surface  bounded  by  the 
closed  gauche  curve  s.  S'  is  a  surface  which  divides  space  into 
two  portions  in  each  of  which  the  components  of  a  vector  Q  are 
represented  by  analytic  functions.  At  S',  some  of  the  com 
ponents  of  Q  parallel  to  the  surface  are  discontinuous.  S'  cuts 
S  in  the  curve  s'  which  divides  S  into  two  portions,  Si  and  Sz. 
Two  curves  in  Sl  and  S2  respectively  drawn  parallel  to  s'  and 
very  close  to  it  shall  be  called  V  ands2'-  Kn  shall  be  the  con 
tinuous  component,  in  the  direction  of  the  normal  to  S,  of  the 
curl  of  Q.  That  portion  of  s  which  with  s/  embraces  practi 
cally  the  whole  of  Si  shall  be  called  st ;  that  portion  of  the 
remainder  of  s  which  with  s.2'  embraces  nearly  the  whole  of 
S2  is  to  be  denoted  by  s2.  Apply  Stokes's  Theorem  to  Si  as 
bounded  by  s1  and  s/  and  to  S2  as  bounded  by  s2  and  s2',  and 
show  that  the  line  integral  of  the  tangential  component  of  Q 
around  s  is  not  in  general  accounted  for  by  the  surface 
integral  of  Kn  over  S,  unless  we  assign  to  Kn  on  s'  a  value 
such  that  its  line  integral  along  the  line  is  finite.  What  is 
this  value  ?  On  page  113  Stokes's  Theorem  is  predicated  only 
of  analytic  vectors.  Justify  the  uses  made  of  the  theorem 
on  page  219  and  in  Sections  82  and  88. 


384  MISCELLANEOUS    PROBLEMS. 

143.  Assuming  that  the  surface  integral  of  the  normal  out 
ward  component  of  any  vector  taken  over  any  closed  surface  S, 
within  and  on  which  the  vector  is  analytic,  is  equal  to  the 
volume  integral  of  the  divergence  of  the  vector  taken  through 
out  the  space  within  the  surface,  show  that  if  in  spherical 
coordinates  R,  0,  3>  are  the  components  of  a  vector  Q,  taken 
in  the  directions  in  which  r,  0,  <f>  increase  most  rapidly,  the 
divergence  of  Q  is  given  by  the  expression 

Dr  (r*R)  •  /  r2  +  De  (sin  0  •  ®)  •  /  r  sin  0  +  D^  •  /  r  sin  0. 

444.  Assuming  that,  if  £,  ^  £  are  three  analytic  functions 
which  define  a  system  of  orthogonal  curvilinear  coordinates, 
and  if  h^  h^  h$  are  the  gradients  of  these  functions,  the  sur 
face  integral,  taken  over  any  closed  surface  S,  of  £7  -cos  (£,  ri) 
(where  U  is  any  function  analytic  within  and  on  S,  and  (£,  ri) 
is  the  angle  between  the  exterior  normal  to  S  at  any  point  on 
S,  and  the  direction  at  that  point,  in  which  £  increases  most 
rapidly)  is  equal  to  the  volume  integral  extended  through  the 
space  enclosed  by  S,  of  /^  •  \  -h^-d  \U  j\  •  ^]  /#£,  show  that,  if 
Q&  QW  Q$  are  ^ne  components  in  the  directions  in  which  £,  r), 
and  £  increase  most  rapidly,  of  an  analytic  vector  Q,  the 
normal  component  of  U  integrated  all  over  S  gives 


Write  down  an  expression  for  the  divergence  of  an  analytic 
vector  in  terms  of  £,  y,  £,  and,  assuming  that  in  the  case  of 
spherical  coordinates  hr  —  1,  he  =  !/?•,  7fy  =  1/r  sin  0,  show 
that  this  yields  the  result  stated  in  the  last  problem. 

145.  Let  P0  be  a  fixed  point  and  P  a  movable  point  in  the 
unlimited  region  T,  without  a  given  surface  S,  and  let  P0P 
be  denoted  by  r.  Show  that  if  a  function  G'  can  be  found 
which  (1)  on  S  has  the  value  —  1/r,  which  (2)  is  harmonic 
at  P0  and  at  every  other  point  of  T,  and  which  (3)  vanishes 
at  infinity  like  the  Newtonian  Potential  Function  of  a  finite 


MISCELLANEOUS    PROBLEMS.  385 

mass,  G'  is  unique.  Show  also  that  if  G  =  G'  +  I/?*,  and 
if  ic  is  any  function  harmonic  in  T,  which  vanishes  at  infinity 
like  a  Newtonian  Potential  Function  and  has  the  value  tc(> 

at  P0,  47rz£'0—  (w-DnGdS,  where  n  represents  an  exterior 

normal  to  S.  Some  writers  call  G  "  Green's  Function  "  for 
the  given  S  and  the  given  P0 ;  others  reserve  this  name 
for  G'.  Attach  a  physical  meaning  to  G.  Define  a  Green's 
Function  for  space  inside  a  closed  surface  S. 

Show  that  if  S  is  a  plane  and  if  r'  is  the  distance  of  P 
from  the  image,  in  the  plane,  of  the  pole  P0,  the  function  G 
is  1/r-l/r'. 

146.  Show    that    the    expression  I    I  2  pl  •  log  (r/r0)  •  dAlt 

where  r0  is  any  constant,  might  be  used  for  the  logarithmic 
potential  function  of  a  columnar  distribution  of  repelling 
matter. 

147.  Show  that  in  general  the  surface  density  of  a  charge 
distributed  on  a  conductor  is   greatest  at  points  where  the 
convex  curvature  of  the  surface  of  the  conductor  is  greatest. 

148.  Show  that  if  Z,  ra,  n  are  scalar  point  functions  which 
define  a  set  of  orthogonal  curvilinear  coordinates  in  an  electric 
field  in  air  where  the  potential  function  is  V,  and  if  L,  M,  N 
represent  the  force  components  taken  at  every  point  in  the 
directions  in  which  the  coordinates   increase   most  rapidly, 
L  =  -  ht  •  Df  V,  M=  -  hm  •  Dm  V,  N=  •*•  hn  •  Dn  V,  and  Laplace's 
Equation  can  be  written 

A  (L  /hm  •  /g  +  DM  (M/ht  •  k»)  +  Dn  (N/h, .  hm)  =  0. 

149.  Prove  that  if  a  distribution  of  electricity  over  a  closed 
surface  produces  a  force  at  every  point  of  the  surface  perpen 
dicular  to  it,  the  potential  function  is  constant  within  the 
surface. 

150.  Two  conducting  spheres  of  radii  6  and  8  respectively 
are  connected  by  a  long  fine  wire,  and  are  supposed  not  to  be 


386  MISCELLANEOUS    PROBLEMS. 

exposed  to  each  other's  influences.  If  a  charge  of  70  units  of 
electricity  be  given  to  the  composite  conductor,  show  that  30 
units  will  go  to  charge  the  smaller  sphere  and  40  units  to  the 
larger  sphere,  if  we  neglect  the  capacity  of  the  wire.  Show 

25 
also  that  the  tension  in  the  case  of  the  smaller  sphere  is  • 

2oO7T 

per  square  unit  of  surface. 

151.  The  first  of  three  conducting  spheres,  A,  B,  and  C,  of 
radii  3,  2,  and  1  respectively,  remote  from  one  another,  is 
charged  to  potential   9.     If  A  be  connected  with  B  for  an 
instant,  by  means  of  a  fine  wire,  and  if  then  B  be  connected 
with  C  in  the  same  way,  C"s  charge  will  be  3  •  6.  [Stone.]    If, 
in  the  last  example,  all  three  conductors  be  connected  at  the 
same  time,  C 's  charge  will  be  4  •  5. 

152.  A  charge  of  M  units  of  electricity  is  communicated  to 
a  composite  conductor  made  up  of  two  widely  separated  ellip 
soidal  conductors,  of  semiaxes  2,  3,  4  and  4,  6,  8  respectively, 
connected  by  a  fine  wire.     Show  that  the  charges  on  the  two 
ellipsoids  will  be  -J  M  and  f  M  respectively.     Compare  the 
values  of  2  Tnr2  at  corresponding  points  of  the  two  conductors. 

153.  Can  two  electrified  bodies  attract  or  repel  each  other 
when  no  lines  of  force  can  be  drawn  from  one  body  to  the 
other  ? 

154.  Two  conductors,  A  and  B,  connected  with  the  earth  are 
exposed  to  the  inductive  action  of  a  third  charged  body.     Do 
A  and  B  act  upon  each  other  ?     If  so,  how  ? 

155.  A  spherical  conductor  A,  of  radius  a,  charged  with  M 
units  of  electricity,  is  surrounded  by  n  conducting  spherical 
shells  concentric  with  it.     Each  shell  is  of  thickness  a,  and  is 
separated  from  its  neighbors  by  empty  spaces  of  thickness  a. 
Show  that  the  limit  approached  by  VA  as  n  is  made  larger  and 
larger  is  (M/a)  log  2. 

156.  The   superficial    density   has   the   same   sign   at   all 
points  of  a  conducting  surface  outside  which  there  is  no  free 
electricity. 


MISCELLANEOUS    PROBLEMS.  387 

157.  An  insulated  and  uncharged  spherical  conductor  of 
radius  4  centimetres  contains  an  eccentric  spherical  cavity 
the  radius  of  which  is  2  centimetres.     At  the  centre  of  the 
cavity  is  a  point  charge  of  10  units.     Show  that  the  charges 
on  the  inner  and  outer  surfaces  are  uniformly  distributed  and 
that  the  value  of  the  potential  function  at  all  points  within 
the  cavity  is  10  /r  —  2.5. 

158.  A  spherical  conductor  of  10  centimetres  radius  is  sur 
rounded  by  a  concentric  conducting  spherical  shell  of  radii 
12  centimetres  and  15  centimetres.    The  sphere  is  at  potential 
zero  and  the  shell  at  potential  unity.     Show  that  the  charges 
are  -  60,  60,  and  15. 

159.  Prove  that  the  electrical  capacity  of  a  conductor  is 
less  than  that  of  any  other  conductor  in  which  it  can  be 
geometrically  enclosed. 

160.  Show  that  two  exactly  similar  conductors  symmetri 
cally  situated  on  opposite  sides  of  a  plane,  so  that  one  is 
the  optical  image  of  the  other  in  the  plane,  repel  each  other 
if  raised  to  the  same  potential. 

161.  Prove  that  the  following  statements  are  true  :  If  any 
conductors,  some  or  all  of  which  are  charged,  are  exposed  to 
one  another's  influences  but  are  far  removed  from  all  other 
charged  bodies,  the  charge  on  one,  at  least,  of  the  conductors 
must  have  the  same  sign  throughout.     If  two  charged  con 
ductors,  uninfluenced  except  by  each  other,  have  equal  and 
opposite  charges,  the  surface  density  at  every  point  of  one 
has  one  sign  and  the  surface  density  at  every  point  of  the 
other  the  opposite  sign.     A  charge,  —  1,  concentrated  at  any 
point  P  produces  a  distribution  of  one  sign  throughout  upon 
a  conductor  C  which  carries  a  total  charge  of  1  +  /x,  /x  being 
any  positive  quantity  whatever.     If  two  conductors  influenced 
only  by  each  other  are  at  potentials  of  the  same  sign,  the 
distribution  has  the  same  sign  throughout  upon  that  one  of 
the  conductors  the  potential  of  which  is  the  greatest  in  abso 
lute  value.     If  two  conductors  influenced  only  by  each  other 


388  MISCELLANEOUS    PROBLEMS. 

are  at  opposite  potentials,  the  distribution  in  each  has  the 
same  sign  everywhere  that  the  potential  function  has.  A 
charged  conductor  is  always  attracted,  in  the  absence  of 
other  charged  bodies,  by  every  other  conductor,  in  its  neigh 
borhood,  which  is  put  to  earth.  [Duheru.] 

If  n  is  the  number  of  unit  Faraday  tubes,  per  square  centi 
metre,  which  pass  through  any  small  portion  of  an  equipoten- 
tial  surface  of  an  electric  field  in  air,  the  strength  of  the  field 
on  this  small  area  is  kirn. 

162.  If  when  a  unit  charge  is  placed  on  a  conductor  C  in 
the  presence  of  conductors  Clt  C2,  kept  at  potential  zero,  the 
charges  on  these  are  —  elt  —  e.2 ;  then  if  C  be  discharged  and 
insulated  and  G\,  C2  be  raised  to  potentials  Vv  F2,  the  potential 
of  C  will  be  <?!?!  + C,  F2. 

163.  A  soap  bubble  of  surface  tension  T  has  a  charge  Q. 
Show  that  its  diameter  is  Q^/(2-n-T)^ 

164.  Prove  that  the  capacity  of  n  equal  spherical  condensers 

when  arranged  in  cascade  is  only  about  -th  of  the  capacity  of 

one  of  the  condensers  ;  but  that  if  the  inner  spheres  of  all  the 
condensers  be  connected  together  by  fine  wires,  and  the  outer 
conductors  be  also  connected  together,  the  capacity  of  the 
complex  condenser  thus  formed  is  about  n  times  that  of  a  single 
one  of  the  condensers. 

165.  A  conductor  the  equation  of  the  surface  of  which  is 

£+j£+*.i 

25      16       9 

is  charged  with  80  units  of  electricity ;  what  is  the  density 
at  a  point  for  which  x  =  3,  y  =  3  ? 

If  the  density  at  this  point  be  a,  what  is  the  whole  charge 
on  the  ellipsoid? 

166.  A  charged  insulated  conductor  A  is  so  surrounded  by 
a  number  of  separate  conductors  B,  (7,  D,  •  ••,  which  are  put 
to  earth,  that  no  perfectly  straight  line  can  be  drawn  from 


MISCELLANEOUS    PROBLEMS.  389 

any  point  of  A  to  the  walls  of  the  room  without  encountering 
one  of  these  other  conductors.  Will  there  be  any  induced 
charge  on  the  walls  of  the  room? 

167.  Assuming  that  in  the  case  of  a  conductor  surrounded 
by  dry  air,  8007r  dynes  per  square  centimetre  is  the  greatest 
pressure  that  a  charge  on  the  conductor  can    exert   at  any 
point   upon  the  air  without  breaking  down  the    insulation, 
show  that  a  spherical  conductor  must  have  a  diameter  of  at 
least  0.126  centimetres  in  order  to  hold,  in  dry  air,  one  elec 
trostatic  unit  of  electricity. 

168.  Prove  that  two  pith  balls  each  4  millimetres  in  diame 
ter  and  3  milligrammes  in  weight,  suspended  side  by  side  by 
vertical  silk  fibres  10  centimetres  long,  cannot  be  so  highly 
charged  with  electricity  that  the  fibres  shall  make  an  angle 
of  60°  with  each  other. 

169.  Discuss  the  following  passage  from  Maxwell's  Elemen 
tary  Treatise  on  Electricity : 

"  Let  it  be  required  to  determine  the  equipotential  surfaces 
due  to  the  electrification  of  the  conductor  C  placed  on  an  insu 
lating  stand.  Connect  the  conductor  with  one  electrode  of  the 
electroscope,  the  other  being  connected  with  the  earth.  Elec 
trify  the  exploring  sphere,*  and,  carrying  it  by  the  insulating 
handle,  bring  its  centre  to  a  given  point.  Connect  the  elec 
trodes  for  an  instant,  and  then  move  the  sphere  in  such  a  path 
that  the  indication  of  the  electroscope  remains  zero.  This 
path  will  lie  on  an  equipotential  surface." 

170.  A  condenser  consists  of  a  sphere  A  of  radius  100  sur 
rounded  by  a  concentric  shell  the  inner  radius  of  which  is  101, 
and  outer  radius  150.    The  shell  is  put  to  earth,  and  the  sphere 
has  a  charge  of  200  units  of  positive  electricity.     A  sphere  B 
of  radius  100  outside  the  condenser  can  be  connected  with  the 
condenser's  sphere  by  means  of  a  fine  insulated  wire  passing 


*  A  very  small  conducting  sphere  fitted  with  an  insulating  handle. 


390  MISCELLANEOUS   PROBLEMS. 

through  a  small  hole  in  the  shell.  B  is  connected  with  A  ; 
the  connection  is  then  broken,  and  B  is  discharged  ;  the  con 
nection  is  then  'made  and  broken  as  before,  and  B  is  again 
discharged.  After  this  process  has  been  gone  through  with 
five  times,  what  is  A's  potential  ?  What  would  it  become  if 
the  shell  were  to  be  removed  without  touching  A  ? 

[2(101)4/(102)5,  2  (101)5/(102)5.] 

171.  If  the  condenser  mentioned  in  the  last  problem  be 
insulated  and  a  charge  of  100  units  of  positive  electricity 
be  given  to  the  shell,  what  will  be  the  potential  of  the  sphere  ? 
of  the  shell  ?     If  we  then  connect  the  sphere  with  the  earth 
by  a  fine  insulated  wire  passing  through  the  shell,  what  will 
be  the  charge  on  the  outside  of  the  shell?     What  will  be  the 
potential  of  the  shell  ?     If  next  A  be  insulated,  and  the  shell 
be  put  to  earth,  what  will  be  A's  potential  ?     What  will  be 
its  potential  if  the  shell  be  now  wholly  removed  ?  VKftf  Wu»  *U 
flat  M^nfcaL'ir  fa-  carlX  te  *&4  tfw^  4m»'e4> 

[2/3,  2/3,  -  4040/41 ;  60/41,  2/205,  - 2/205,  -  202  /205.] 

172.  A  conductor  is  charged  by  repeated  contacts  with  a 
plate  which  after  each  contact  is  recharged  with  a  quantity 
(E)  of  electricity  from  an  electrophorus.     Prove  that  if  e  is 
the  charge   of   the  conductor  after  the    first   operation,  the 
ultimate  charge  is  E  e  /  (E  —  e). 

173.  If  one  of  a  system  of  n  conductors  entirely  surrounds 
all  the  others,  2  n  —  1  of  the  coefficients  of  potential  have  the 
common  value  p.     If  the  outside  conductor  be  put  to  earth,  it 
loses  a  quantity  Q  of  electricity.     Show  that  the  energy  loss 
is  \pQ\ 

174.  A  conductor  is  formed  of  two  infinite  planes  inter 
secting  at  right  angles  and  is  connected  with  the  earth.     A 
long  straight  wire,  parallel  to  the  intersection  of  the  planes, 
at  distances  b  and  a  from  them,   has   an  electric  charge  e 
per  unit  length.     Show  that  the  electrification  of  the  first 


MISCELLANEOUS    PROBLEMS.  391 

plane  at  a  distance  x  from  the  line  of  intersection  of  the 

-  4  abex/7r [(a2  +  b-  +  x2)2  -  4  aV]. 

175.  The  energy,  per  unit  of  surface,  of  a  plane  parallel 
plate  condenser  in  which  the  superficial  charge  density  is 
<TO  is  2  7ro-02a  when  the  distance  between  the  plates  is  a.     Show 
that  if  the  distance  be  decreased  to  a  —  Aa  the  energy  is 

27ro-02(>-  Aa) 
if  the  charge  remains  constant,  and 

2  mr0*a*  /  (a  -  Aa) 

if  the  potential  remains  constant.  Hence  prove  that  the  rates 
of  change  of  the  energy  are  equal  in  value  but  opposite  in  sign 
in  the  two  cases. 

1 76.  The  foot  of  the  perpendicular  dropped  from  any  point  P 
upon  the  line  A^AZ  shall  be  marked  J/.     At  A^  is  a  point  charge 
ml  and  at  A2  a  point  charge  —  ra2,  m±  being  greater  than 
^P  =  rv  A2P  =  rv  AJf  =  x,  MP  =  y,  A,A2  =  a,  rn^/m^ 
Show  that  the  surface  integral  of  normal  force  parallel  to 
the  x  axis  over  an  infinite  plane  through  M  perpendicular 
to  A^A2  is  2 TT (ml  —  m.2)  if  x  >a ;  2  TT  (?nl  +  m2)  if  0  <  x  <  a  ; 
and  2  TT  (m2  —  m^  if  a;  <  0.     The  induction  outward  through 
an  infinite  spherical  surface  with  centre  at  any  finite  point  is 
4  TT  (ml  —  ra2).     Show  that  the  value  at  any  point  on  a  spherical 
surface  of  radius  r^  with  centre  at  Al}  of  the  normal  outward 
component  of  the  force  is  rn^/r-f  —  ra2  cos  (r15  r2)/r22,  and  this 
is  positive  for  every  point  of  the  surface  if  rl>a,fji/(p  —  1). 
It  follows  from  this  that  no  line  of  force  can  come  from 
infinity  to  the  charge  on  A«\  but  47r^2  of  the  4  irmv  lines 
which  start  from  A^  reach  A2.     Show  that  all  the  lines   of 
force  which   cross  the  two  planes  drawn   perpendicular  to 
A^  through  At  and  A2  cross  them  from  left  to  right.     The 
inductions  across  these  planes  are  2irmz  and  2-irm^     Through 
M}  any  point  of  A^A^  imagine  a  plane  drawn  perpendicular  to 


392  MISCELLANEOUS    PROBLEMS. 

A-iAt  and  let  a  circumference  be  drawn  on  this  plane  with  If  as 
centre  and  MR  as  radius.  Let  the  angles  which  A^R  and  A2R 
make  with  the  line  from  Al  to  A2  be  o^  and  o>2,  then  the  induc 
tion  through  the  circle  is  27^?%  (1  —  cos  o^)  -f  m2(cos  o>2  —  1)] 
or  2ir[ml(L  —  cos  o^)  +  ina  (1  +  cos  o>2)]  according  as  A^M  is 
greater  than  a,  or  positive,  and  less  than  a.  If  in  the  last  case 
the  radius  be  so  chosen  that  the  circle  shall  include  all  the 


FIG.  125. 

lines  which  converge  to  A2,  we  must  equate  the  induction  to 
4  Trm2.  This  yields  mL  cos  o^  —  m2  cos  o>2  =  ml  —  m2,  which  may 
be  regarded  as  the  equation  of  the  surface  of  separation 
between  the  lines  which  go  from  Al  to  A2  and  those  which  go 
to  infinity. 

2  TT  [m:  (1  —  cos  wj)  +  wi2  (1  +  cos  o>2)]  =  (7  is  the  equation  of 
a  surface  of  revolution  which  includes  everywhere   C  lines 


MISCELLANEOUS    PROBLEMS.  393 

of  force.  Since  every  meridian  curve  of  this  surface  is  itself 
a  line  of  force,  the  equation  just  written  may  be  regarded  as 
the  general  equation  of  the  lines  of  force.  If  mv  =  m2,  the 
lines  are  sometimes  called  "magnetic  lines."  In  this  case 
the  equation  becomes  cos  o^  —  cos  w2  =  const.,  and  the  lines 
have  the  forms  of  the  curves  which  pass  through  the  points 
N,  S  in  Fig.  125. 

Show  that  if  p.  =  1,  if  R  is  the  resultant  force  at  any  point 
P,  and  if  Q  is  the  point  where  the  line  of  action  of  R  cuts  A^A^ 
jK/[sin  (rlt  r2)]  =  ??i/[r12  sin  (It,  >-2)]  =  w/[r22  sin  (R,  r^~\  or, 


since  sin  (A,PQ)  =  (sin  PQA 

and  sin  (  A2PQ)  =  (sin  PQAJ  (QA2)  /  r2, 


If  Q  is  fixed,  P  must  move  so  that  i\/r.2  is  constant  :  its  locus 
is,  therefore,  a  circle.  [See  Mascart  et  Joubert,  §§  168  and 
169,  and  also  Nipher's  Electricity  and  Magnetism,  Ch.  III.] 

177.  Two  condensers  A  and  B  have  capacities  Ci  and  C2. 
A  is  charged  by  a  certain  battery  and  then  discharged  ;  it  is 
then  charged  and  its  charge  is  shared  with  B  ;  finally  A  and  B 
are  both  discharged.  Show  that  the  energies  of  the  different 
discharges  are  to  each  other  as 


[Clare  College.] 

178.  An  earth-connected  circular  disc  5  centimetres  in  radius 
is  suspended  horizontally  from  one  arm  of  a  balance,  and  an 
insulated  plate  is  placed  parallel  to  it  and  1  centimetre  below 
it.  When  the  lower  plate  has  been  electrified  there  is  found 
to  be  an  attraction  equal  to  the  weight  of  1.274  grammes 
between  the  two.  Show  that,  assuming  the  electricity  to 
be  uniformly  distributed,  the  potential  of  the  lower  plate  is 
about  6000  volts. 


394  MISCELLANEOUS    PROBLEMS. 

179.  S  is  an  equipotential  surface  due  to  a  distribution  of 
matter  of  which  it  encloses  a  portion  Mv  and  excludes  a  por 
tion  M2.     Let  Ml  be  distributed  on  S  according  to  the  law 
4  TTCT  =  —  DnV:  then  superpose  on  the  system  thus  formed  the 
negative  of  the  original  system,  so  as  to  have  the  surface  S  at 
zero  potential  due  to  the  distribution  on  it  and  to  the  negative 
of  MI  within  it.     What  will  now  be  the  value  of  the  poten 
tial  function  without  S?     At  a  distance  82  from  the  centre 
of  a  spherical  cavity  of  radius  r,  in  a  conductor  which  is  at 
potential  zero,  is  a  point  charge  of  m2  units.     Find  by  aid  of 
the  formulas  given  in  Section  65  the  density  of  the  charge 
on  the  wall  of  the  cavity. 

180.  If  a  conductor  C,  which  entirely  surrounds  a  system 
of  charged  and  insulated  conductors,  be  at  first  insulated  and 
at  potential  V,  and  then  put  to  earth,  the  potentials  of  all  the 
interior  conductors  will  be  diminished  by  V.     If  this  system 
be  now  discharged,  the  loss  of  energy  is  the  same  as  if  C  had 
not  been  put  to  earth  but  had  had  the  interior  conductors  put 
into  connection  with  its  inner  surface.     [M.  T.] 

181.  Show  that  r/Sths  of  the  unit  Faraday  tubes  proceeding 
from  an  electrified  particle,  at  a  distance  8  from  the  centre  of 
a  conducting  sphere  of  radius  r,  which  is  put  to  earth,  meet 
the  sphere,  if  there  are  no  other  conductors  in  the  neighbor 
hood,  and  that  the  rest  go  off  to  infinity. 

182.  If  a  charge  ml  is  placed  at  a  point  Al  distant  8X  from 
the  centre  0  of  a  conducting  sphere  of  radius  r  (Section  65) 
kept  at  potential  zero,  the  charge  induced  on  the  surface  has 

v  the  density  cr  =  —  m1  (8^  —  r2)/47rrr13  at  a  point  distant  r} 
from  AV  and  the  total  amount  of  the  induced  charge  is 
—  ra^/Sp  The  attraction  between  the  point  charge  and  the 
induced  charge  is  m^rSl/(8]2  —  r2)2.  If  now  a  charge  M  be 
distributed  uniformly  over  the  sphere  so  as  to  raise  its 
potential  to  M/r  or  F0,  the  new  density  will  be 

-  m,  (V  -  r2)]  /4  TT  rrf, 


MISCELLANEOUS    PROBLEMS.  395 

the  new  charge  E  =  VQr  —  »?i>'/Si,  and  the  attraction 
F  =  [<>•  8i/  (8f  -  r)2  -  m,  F0  r/&fi, 

or  m*r  ^/(Sf  -  f2)2-  m.E/S'2  -  <  r/Sf. 

« 
This  attraction  is  zero  when  8L  satisfies  the  equation 

E  B,  (8f  -  r2) 2  =  m,  r3  (2  8^  -  r5). 

If  M=+mlr/8l,  the  total  charge  on  the  sphere  will  be 
zero.  In  this  case  F0  =  ^1/81,  and  the  force  of  attraction  is 
w12^(2812  —  »*)/(8i2-  r2)2^3:  this  expression  is  always  posi 
tive.  The  density  on  the  sphere  is  zero,  if  anywhere,  on  a 
circumference  determined  by  the  equation 

E  +  ?H!  T/S!  =  ™i  r(^  ~  ^)/r?. 

0  and  A2  are  inverse  points  with  respect  to  a  spherical 
surface  S  of  radius  VSf  —  r2,  the  centre  of  which  is  ^.  If, 
therefore,  T  is  any  point  on  S,  A.2T-Bl=  OT-  VSL2  —  r2  and, 
if  Jf  =  mx  r  j  VSf  —  -r,  the  potential  function  has  the  same 
uniform  value  on  *S  and  on  the  conductor.  The  intersection 
of  the  two  surfaces  is  a  line  of  no  force  and  no  density. 

The  potential  function  due  to  ml  alone  is  the  same  as  that 
due  to  ???j  and  the  charged  sphere,  at  all  points  on  the  spherical 
surface  OP  j  A.2P  =  M  8X  /ml  r :  if  E=  0,  this  is  the  plane  which 
bisects  A20  at  right  angles. 

The  mutual  potential  energy  of  the  point  charge  m^  and  the 
distribution  on  the  sphere  is 

-  i  »w12rs/812(812  -  >~). 

Show  that  if  a  charged  conducting  sphere  of  radius  10  centi 
metres  is  at  potential  F0  in  the  presence  of  a  point  charge  of 
12  units  at  a  distance  of  20  centimetres  from  the  centre  of 
the  sphere,  the  whole  charge  on  the  sphere  is  10  (FJ,  —  f). 


396  MISCELLANEOUS    PROBLEMS. 

Show  also  that  F0  is  2/15,  6/5V3,  or  3.6  according  as  the 
density  of  the  surface  charge  is  zero,  at  the  point  of  the  sur 
face  farthest  from  AI,  at  a  point  just  visible  from  A^  or 
at  the  point  nearest  AI.  Show  that  if  the  whole  charge  011 
•»the  spherical  surface  is  14/3  there  is  no  attraction  between 
the  point  charge  and  the  surface  charge ;  and  that  if  the 
sphere  was  originally  uncharged  and  insulated,  its  potential 
was  constantly  equal  to  12  /8X  as  the  point  charge  gradually 
approached  its  present  position  from  infinity. 

Show  that  the  integral  of  (Sf  —  r2)/^3  taken  over  the  sur 
face  of  the  sphere  is  4  Trr2/^.  How  much  of  the  charge  on 
the  sphere  is  visible  from  Av  ? 

Find  the  surface  density  on  a  spherical  conductor  at  poten 
tial  zero  under  the  action  of  two  equal  external  point  charges 
situated  at  equal  distances  on  opposite  sides  of  the  centre. 
Consider  separately  the  case  where  the  point  charges  have 
opposite  signs. 

183.  An  insulated  conducting  sphere  of  radius  r  charged 
with  m  units  of  positive  electricity  is  influenced  by  m  units 
of  positive  electricity  concentrated  at  a  point  2r  distant  from 
the  centre  of  the  sphere.     Give  approximately  the  general 
shape  of  the  equipotential  surfaces  in  the  neighborhood  of 
the  sphere. 

Give  an  instance  of  a  positively  electrified  body  the  poten 
tial  of  which  is  negative. 

184.  Prove  that  if  the  spherical  surfaces  of  radii  a  and  b, 
which  form  a  spherical  condenser,  are  made  slightly  eccentric, 
c  being  the  distance  between  their  centres,  the  change  of  elec- 

3  abc -  cos  0 

trification  at  any  point  of  either  surface  is  - — -77, r-> 

47r(&  —  a)(b3  —  as) 

where  0  is  the  angular  distance  of  the  point  from  the  line  of 
centres,  and  where  the  difference  between  the  values  of  the 
potential  function  on  the  two  surfaces  is  unity. 

185.  Show  that  if  an  insulated  conducting  sphere  of  radius  a 
be  placed  in  a  region  of  uniform  force  (^o)?  acting  parallel  to 


MISCELLANEOUS    PKOBLEMS.  397 

the  axis  of  x,  the  function  —  XQ  •  x  (1  —  a3/?-3)  +  C  satisfies  all 
the  conditions  which  the  potential  function  outside  the  sphere 
must  satisfy,  and  is  therefore  itself  the  potential  function. 
Show  that  the  surface  density  of  the  charge  on  the  sphere 

is  —    —j  and  prove  that  this  result  might  have  been  obtained  by 
4  ird 

making  ^  infinite  in  the  formulas  near  the  top  of  page  206. 

186.  If  <7n,  </22  are  the  coefficients  of  capacity  of  two  of  a 
set  of  conductors,  and  if  ^12  is  their  coefficient  of  mutual 
induction,  the  capacity  of  the  compound  conductor  formed  by 
joining  these  two  conductors  by  a  fine  wire  is  qn  +  2  qlz  +  qw 
if  all  the  other  conductors  be  put  to  earth.  If  pn,  p^,  plz 
are  the  coefficients  of  potential  of  the  two  conductors,  and 
if  all  the  other  conductors  of  the  series  are  uncharged  and 
insulated,  the  capacity  of  the  compound  conductor  is 


If  the  distance  b  between  the  centres  of  two  conducting 
spheres  of  radii  av  a2  is  large  compared  with  the  diameter  of 
either,  pu  =  1/alf  p.22  =  l/«2,  and  _p12  is  approximately  1/6, 
so  that  if  elt  e2  are  the  charges  of  the  spheres  and  Vlf  Vz  their 
potentials,  Fx  =  el/al  +  e.,/b,  V2  =  el/b  +  e.2/a.2.  Show  that, 
approximately, 


q.2.2  =  ajr  /  (Ir  —  a^a^). 

187.  If  on  the  radius  vector  OP  drawn  from  a  fixed  point 
0  to  another  point  P  a  new  point  P'  be  taken,  such  that 
OP  •  OP'  =  a2,  where  a  is  a  constant  chosen  at  pleasure, 
P  and  P'  are  said  to  be  inverse  points,  0  is  the  centre  of 
inversion,  a  sphere  of  radius  a  with  centre  at  0  is  the  sphere 
of  inversion  and  a  the  radius  of  inversion.  One  of  a  pair  of 
inverse  points  is  without  the  sphere  of  inversion  and  the 
other  within,  unless  both  coincide,  The  straight  line  which 


398  MISCELLANEOUS    PROBLEMS. 

joins  the  points  of  contact  of  tangents  to  the  sphere  drawn 
from  an  outside  point  P'  passes  through  the  inverse  point  P. 
If  P,  P'  and  Q,  Q'  are  pairs  of  inverse  points,  the  triangles 
OPQ  and  OQ'P'  are  similar.  If  one  (P)  of  a  pair  of  inverse 
points  moves  along  a  curve,  or  over  a  surface,  or  through  a 
space,  the  other  (P1)  will  generate  the  inverse  curve,  surface, 
or  space.  A  plane  at  a  perpendicular  distance  b  from  0 
inverts  into  a  spherical  surface  of  radius  a2/ 2  b,  passing 
through  0.  A  spherical  surface  of  radius  c  with  centre  at  a 
distance  b  from  0  inverts  into  another  spherical  surface  of 
radius  a2c/(b2  —  c2)  with  centre  at  a  distance  a2b/(b2  —  c2) 
from  0.  If  a2  =  b2  —  c2,  the  spherical  surface  inverts  into 
itself,  though  the  inverse  of  the  old  centre  is  not  the  new 
centre.  The  centre  of  inversion  inverts  into  the  region  at 
infinity. 

Prove  that  if  the  origin  be  the  centre  of  inversion,  a  point 
P  or  (x,  y,  TV),  distant  r  from  the  origin,  inverts  into  a  point 
P'  or  (a;',  y'j  z'),  distant  r'  from  the  origin,  where  rr'  =  a?, 
x/r  =  x'/r'j  y /r  =  y'/r',  z/r  =  z'/r',  x  =  a2x'/r12,  y  =  a2y'/r'2, 
z  =  a2z'/r'2,  x'  =  a2x/r*,  y'  =  a?y/r>,  z'  =  a?z/r>.  An  element 
of  arc  ds  at  P  inverts  into  an  element  of  arc  ds'  at  P',  such 
that  ds  =  r2 -ds' /a2  =  a2 -ds' /r12.  An  element  of  area  dS  at 
P  inverts  into  an  element  of  area  dS1  at  P',  such  that 
dS=i*-dS'/a4  =  a4>dS'/r'*.  An  element  of  volume  dr  at 
P  inverts  into  an  element  of  volume  dr'  at  P',  such  that 
dr  =  rQ-dr'/afi  =  aP-dr'/r'6.  The  angle  between  two  curves 
which  intersect  at  P  is  equal  to  the  angle  between  the  inverse 
curves  which  intersect  at  P'.  If  P  and  P'  be  drawn  in 
different  diagrams,  in  which  the  rectangular  Cartesian  coordi 
nates  are  x,  y,  z  and  x',  y',  z'  respectively,  £  =  a2x' /r'2, 
y  =  a2y' /r'2,  £  =  a?z' /r'2,  define  a  set  of  orthogonal  curvi 
linear  coordinates  in  the  second  diagram,  and  the  Cartesian 
coordinates  of  P  in  the  first  diagram  are  equal  to  the  curvi 
linear  coordinates  of  P'  in  the  second  diagram.  Any  func 
tion  F(x,  y,  z)  has  the  same  numerical  value  at  P  that  the 


MISCELLANEOUS    PROBLEMS.  399 

function  F(a*x'/r'*,  a*y'/r'*,  aV/V2)  =  f  (xf,  y',  z')  has  at  P'. 
Prove  that 

(D?  +  #/  +  A2)  ^  (*,  y,  «)  at  P 

=  (r'YO  (A-2  +  D/  +  A-2)  KA')  atp-. 

If  P  is  zero  on  any  surface  or  throughout  any  space  in  the 
first  diagram,  aif//r'  is  zero  on  the  corresponding  surface  or 
throughout  the  corresponding  space.  If  F  has  the  constant 
value  c  on  the  surface  S,  cnf//r'  has  the  value  ac/r',  which  is 
not  constant  on  the  corresponding  surface  S'. 

If  F  is  the  potential  function  due  to  a  volume  distribution 
of  density  p  in  a  region  T,  together  with  a  superficial  distri 
bution  of  density  a-  on  a  surface  S  and  a  point  charge  e  at  a 
point  (),  (aiA/7'')  is  th®  potential  function  due  to  a  volume 
distribution  of  density  p'  =  a5p/r'5  in  the  region  Tf,  corre 
sponding  to  T,  together  with  a  superficial  distribution  of 
density  o-'  =  <z3cr/r'3  on  the  surface  S',  which  corresponds  to  S, 
and  a  point  charge  e'  =  r'e/a  at  the  point  Q',  which  is  the 
inverse  of  Q.  The  inverse  of  a  point  charge  e  at  the  centre 
of  inversion  is  a  charge  at  infinity,  which  raises  all  finite 
points  to  potential  e/a.  If  F  is  the  potential  function  of  a 
distribution  o-,  p  which  keeps  a  certain  surface  S  at  potential 
zero,  (a\f//r')  will  be  the  potential  function  of  a  distribution 
a-',  p'  which  keeps  the  corresponding  surface  S'  at  potential 
zero.  If  F  is  the  potential  function  of  a  distribution  o-,  p 
which  keeps  the  surface  S  at  potential  c,  (a^//r')  will  be  the 
potential  function  of  a  distribution  or',  p'  which  keeps  S'  at 
the  potential  ac/r':  if,  however,  we  add  to  the  distribution 
o-',  p'  a  point  charge  —  ac  at  the  origin,  the  new  potential 
function  will  keep  S'  at  potential  zero. 

188.  Show  that  if  a  point  charge  e  be  anywhere  between 
two  infinite  planes  which  form  a  diedral  angle  of  60°,  these 
planes  would  form  a  surface  of  potential  zero  due  to  the 
original  charge  and  five  images  in  the  planes.  Find  the  den 
sity  of  the  charge  on  two  planes  which  form  an  angle  TT/W, 


400  MISCELLANEOUS    PROBLEMS. 

if  they  are  kept  at  potential  zero  in  presence  of  a  point 
charge  between  them,  invert  the  system  with  respect  to  the 
charged  point. 

189.  A  homogeneous  sphere  of  density  p  and  radius  c  has 
its  centre  at  a  point  C  distant  d  from  an  outside  point  0. 
The  value  of  the  potential  function  at  a  point  P  outside  the 
sphere  is  $wp&/CP.     Show  that  if  the  distribution  be  inverted, 
using  0  as  centre,  the  new  distribution  is  a  heterogeneous,  cen- 
trobaric  sphere  of  mass  %irp<?a,/d,  the  baric  centre  of  which 
is  the  inverse  point  of  C.     [Routh.] 

190.  A  point  charge  -f-  e  lies  on  the  x  axis  at  a  distance  +  b 
from  the  origin  between  two  conducting  plates,  x  =  0,  x  =  2  c, 
both  of  which  are  kept  at  zero.     Show  that  the  images  of  the 
point  charge   in  the  planes   are   an   infinite   series  of  point 
charges  numerically  equal  to  e  but  alternately  positive  and 
negative  at  points  on  the  x  axis.     The  coordinates  of  the  nega 
tive  images  are  —  I,  —  (4c  +  ft),  —  (8c  +  ft),  •  •  -,(4c  —  b),  (8c  —  b), 
(12  c  —  ft), . . .,  and  those  of  the  positive  images  are  (4  c  -f-  ft), 
(8  c  +  ft),  (12  c  +  ft),  •  •  • ,  -  (4 c  -  ft),  -  (8  c  -  b) ,  -  (12  c  -  ft),  - .  - . 
Show  that  the  force  at  any  point  between  the  planes  might  be 
computed  from  these  images  and  the  original  point  charge. 
Indicate  a  method  for  determining  the  density  of  the  induced 
charges  on  the  plates.     State  clearly  the  result  of  inverting- 
the  system,  using  the  original  charged  point  as  centre  of  inver 
sion,  and  each  of  several  different  values  for  a. 

If  in  this  problem  the  charge  e  is  at  a  point  0  midway 
between  the  plates,  and  if  this  point  be  chosen  as  origin,  ft  =  c, 
and  there  are  positive  images  at  points  the  x  coordinates  of 
which  are  0,  4  c,  8  c,  12  c,  •  •  • ,  —  4  c,  —  8  c,  —  12  c,  •  •  • ,  and  nega 
tive  images  at  points  the  x  coordinates  of  which  are  —  2  c, 
—  6  c,  —  10  c,  •  •  •,  2  c,  6  c,  10  c,  •  •  • .  Show  that  if  the  system 
be  inverted,  using  0  as  centre  of  inversion  and  c  as  radius  of 
inversion,  and  if  the  inverse  of  the  charge  at  0  be  omitted, 
the  result  is  a  conductor  formed  of  two  spherical  surfaces 
of  radius  r  =  $c,  in  contact  at  potential  F"0  =  —  e/c  under 


MISCELLANEOUS    PROBLEMS.  401 

positive  charges  equal  respectively  to  £  e,  |  e,  -fa  e,  •  •  • ,  at 
points  the  x  coordinates  of  which  are  ±  J-  c,  ±  |  c,  ±  TT5  c,  •  •  • ; 
and  negative  charges  \  e,  ^  e,  TV  e>  "  '  >  a^  P°in^s  the  x  coordi 
nates  of  which  are  ±  i  c,  ±  £  c,  ±  y1^  c,  •  •  • .  The  total  charge 
in  each  of  the  spheres  is 

-  i«  (1  -  I  +  4  -  i  •••)  =  -  le  •  log  2  =  [>  •  log  2, 
and  their  mutual  repulsion,  £  F02  (log  2  —  i). 

191.  If  two  spherical  conductors  each  of   radius  a  have 
charges  e^  e2»  and  are  at  a  great  distance  apart,  the  energy  of 
the  system  is  (e?  -f  «22)/2  a.     If  the  two  are  brought  up  into 
contact,  the  whole  charge  of  the  compound  conductor  thus 
formed  is  (el  +  e,2),  it  is  at  potential  (el  +  e.2)  /  2  a  •  log  2,  and 
the  energy  of  the  system  is  (^  +  e2)2/4a-log2.     Show  that 
the  work  done  against  the  mutual  repulsions  of  the  two  charges 
during  the  approach  of  the  spheres  is  about 

[(0.722)6^  -  (0.139)  (ef  +  e.^/a, 
and  discuss  separately  the  special  cases 

el  =  0,  el  =  e.,,  e^  =  5  &,,  e±  =  |  e* 

192.  Show  that  if  a  point  charge  be  situated  at  a  point  0, 
between  two  concentric  spherical  surfaces,  it  is  possible  to 
find  a  series  of  electric  images  which  together  with  the  origi 
nal  charge  would  keep  each  of  the  surfaces  at  potential  zero. 
What  would  be  the  result  of  inverting  the  system,  using  O  as 
centre  ? 

193.  A  certain  condenser  consists  of  a  closed  conducting 
surface  Sl  surrounded  by  another  closed  conducting  surface  $>, 
separated  from  the  first  by  a  homogeneous  dielectric.     When 
the  condenser  is  charged,  the  lines  of  force  between  Sl  and  S.> 
are  the  same  as   if  S2  were  removed  and  ^  freely  charged. 
What  do  you  know  about  Sl  and  $2  ? 

194.  The  semiaxes  of  a  conducting  prolate   ellipsoid   of 
revolution  are  10,  8,  8.    Find,  by  help  of  the  formulas  of  Sec 
tions  6  and  23,  the  external  field  when  the  conductor  has  a 


402  MISCELLANEOUS    PROBLEMS. 

free  charge  of  60  units,  and  show  that  the  surface  density  at 
the  equator  is  then  3  /  16  IT. 

195.  A  prolate  conducting  spheroid  of  major  axis  2  a  and 
minor  axis  2  b,  has  a  charge  of  electricity  E.     Prove  that  the 
attraction  between  the  two  halves  into  which  it  is  divided  by 
its  diametral  plane  is  E~  •  log(a/b  )/4(a2  -  b2).     [St.  John's 
College.] 

196.  If  a  particle  charged  with  a  quantity  e  of  electricity 
be  placed  at  the  middle  point  of  the  line  joining  the  centres 
of  two  equal  spherical  conductors  kept  at  zero  potential,  the 
charge  induced  on  each  sphere  is 

-  2  em  I  -ra      2??t2 


where  m  is  the  ratio  of  the  radius  of  either  of  the  spheres 
to  the  distance  between  their  centres. 

197.  A  conducting  sphere  of  small  radius  a  is  situated  in 
the  open  air  at  a  considerable  height  h  above  the  ground. 
Show  that  its  electrical  capacity  is  increased  by  the  neighbor 


hood  of  the  ground  in  the  ratio  of  1  +     -  —  J  to  1,  very  nearly. 

198.  A  negative  point  charge,  —  e.2,  lies  between  two  posi 
tive  point  charges  el  and  e3  on  the  line  joining  them  and  at 
distances  a  and  b  from  them  respectively.  Show  that  if 


where     1  <  X2  < 


b       a       a  -f  b 

there  is  a  circumference  at  every  point  of  which  the  force 
vanishes. 

199.  Two  spherical  conducting  surfaces  of  radii  a  and  b 
form  a  condenser.  Prove  that  if  the  centres  be  separated 
by  a  small  distance  d,  the  capacity  is  approximately 

ab      C  abd2 


b-a  I          (b-a)(b3 
When  d  =  0,  the  capacity  is 


•} 

-  a3)  J 


MISCELLANEOUS    PROBLEMS.  403 

200.  A  small  insulated  conductor,  originally  uncharged,  is 
connected  alternately  with  two  insulated  conductors  A  and  B 
at  a  considerable  distance  apart.  Prove  that  if  e0  and  e0'  are 
the  original  charges  of  A  and  B,  el  and  e^  their  charges  after 
the  carrier  has  touched  A  and  then  touched  B,  the  charge  of  B 
when  the  carrier  has  touched  A  and  B  each  n  times  is 

ab  —  ej  -  e0'  e0'  —  <?/ 


ab-1  (ab  -  1)  an-lb*-1 

where  a  =  ej  e^  and  b  =  (e0  +  e0'  —  ej  /e^. 

The  charges  of  &  B,  and  the  carrier,  are  ultimately  in 
the  ratios 


201.  If   a   series  of   conductors    were    constructed   which 
might   be  made    to   coincide   with  the    closed  level   surface 
of  a  harmonic  function  w  which  vanishes  at  infinity  like  a 
Newtonian  Potential  Function,  the  capacities  of  any  two  of 
these  conductors  would  be  to  each  other  in  the  ratio  of  the 
reciprocals  of  the  values  of  w  on  the  corresponding  surfaces. 
If  two  of  the  surfaces  for  which  iv  =  wl  and  w  =  w2  <  u\  be 
constructed  of  metal,  and  if  charges  El  and  E2  be  given  them, 
the  energy  is 

2\  '  cc9   '  H — l~cT^ 

where   Ci  and  C2  are  the    capacities.      The  energy  becomes 
\(El  +  E2)2/C2  if  the  two  are  connected. 

202.  An  insulated  conducting  sphere  of  radius  r,  bearing  a 
charge  ra,  is  introduced  into  a  field  of  force  due  to  a  fixed 
distribution  M  of  electricity.     Show  that  if  the  value  of  the 
potential  function  due  to  M  at  the  centre  of  the  sphere  is 
(7,  the  value  of  the  potential  function  within  the  sphere  is 
C  +  m/r. 

203.  Compute  the  force  at  the  point  (#,  ?/,  z)  due  to  a  par 
ticle  of  mass  —  ra  at  the  point  (a,  0,  0)  and  a  particle  of  mass 


404  MISCELLANEOUS    PROBLEMS. 

+  rn  at  the  point  (—  a,  0,  0),  and  show  that  if  m  and  a  be 
made  to  increase  indefinitely  in  such  a  manner  that  the  ratio 
of  m  to  a2  is  always  equal  to  the  constant  A,  the  field  becomes 
ultimately  a  uniform  field  of  intensity  X  =  2  A. 

204.  It  is  evident  that  the  value  at  any  point  P  in  the  xy 
plane,  of  the  potential  function  due  to  two  slender,  infinitely 
long,  homogeneous,  straight  filaments  (of  mass  m  and  —  m 
respectively  per  unit  length)  which  cut  the  plane  perpendicu 
larly  at  the  points  A  and  B,  is  2  m  log  (AP/BP),  and  that  the 
equipotential  surfaces  are  circular  cylinders  [one  is  a  plane] 
such  that  A  and  B  are  inverse  points  with  respect  to  every 
one  of  them.  If  the  radius  of  any  one  of  these  cylindrical 
surfaces  the  axis  of  which  cuts  the  xy  plane  at  C  be  denoted 
by  r  (Fig.  126),  and  if  AP  =  rlt  BP  =  r2,  AC  =  81?  BC  =  32, 

8l/r=r/82  =  rl/r2,  and 
the  triangles  BCP  and 
ACP  are  similar  if  P 
lies  on  the  cylinder.  The 
resultant  force  F  at  P 
has  the  direction  of  the 
normal  to  the  cylinder, 
the  repulsion  due  to  the 
126.  filament  which  cuts  the 

plane  at  A  is  2  m/rL,and 

the  attraction  due  to  the  filament  which  cuts  the  plane  at 
B  is  2m/r2.  If  the  angle  APB  be  denoted  by  a,  the  Principle 
of  the  Parallelogram  of  forces  applied  at  P  yields 

F/ sin  a  =  2  m  /  [r2  sin  (r,  ?-,) ]  =  2  m /  [^  sin  (>•,  r2) ] , 
and  the  Theorem  of  Sines  applied  to  the  triangle  APB  yields 

AB /sin  a  =  7-i/sin  (r,  r^)  =  r2/sin  (r,  r2), 
so  that 

F=  2  mAB/r^  -r2  =  2  m^AB/r  •  ^2  =  2  m^AB/r-  r* 
?  =  2m(r2  -  S22)/r-r2. 


MISCELLANEOUS    PROBLEMS.  405 

The  value  of  the  potential  function  on  the  cylinder  is 
2  m  log  8,  /  /•  =  2  m  log  r  /  82. 

If,  now,  the  mass  of  the  filament  which  cuts  the  plane 
at  B  be  spread  on  the  cylindrical  surface  so  that  the  surface 
density  at  every  point  is 

the  potential  function  outside  the  cylinder  will  be  unchanged. 
If,  finally,  a  mass  m'  per  unit  length  be  spread  uniformly  over 
the  cylinder,  the  value  of  the  potential  function  within 
and  on  the  surface  will  be  2  m  log  (Sj  /  r)  -f-  2  m'  log  r,  and,  by 
a  suitable  choice  of  m',  this  may  be  given  any  value.  The 
whole  charge  on  the  unit  length  of  the  cylindrical  surface  is 
m'  —  m  =  M,  the  value  of  the  potential  function  on  the  sur 
face  is  Vs=2  m  log-^/r)  -+-  2(M  +  m)  log  r,  and  the  surface 
density  at  a  point  distant  r^  from  the  straight  line  which  cuts 
the  paper  perpendicularly  at  A  is 

(M  +  m)  /  2  TT>-  -  m  (V  -  r2)  /  2  TT>-  •  r*. 

At  any  point  Q  without  the  cylinder  the  value  of  the  potential 

function  is 

2m  log  (AQ/BQ)  +  2(J/+  m)  log  CQ. 

Show  that  the  force  of  attraction  between  the  charge  on  the 
cylinder  and  the  unit  length  of  the  filament  through  A  is 

2  m  [m8L  /  (V  -  r2)  -  (M  +  m)  /  8J. 


This  force  vanishes  if  V/'"'2  =  (M  +  w)/^-  Show  also  that 
if  the  cylinder  is  at  potential  zero  in  the  presence  of  the  fila 
ment  through  A,  3f=  —  m  log  (S^r). 

If  we  superpose  upon  this  distribution  a  second  consisting  of 
a  homogeneous  filament  of  mass  —  m  per  unit  of  length  and 
cutting  the  paper  perpendicularly  at  A  and  a  similar  filament 
of  mass  +  m  per  unit  length  cutting  the  paper  at  B,  and  notice 


406  MISCELLANEOUS    PKO13LEMS. 

that  the  potential  function  due  to  the  new  distribution  has 
the  value  —  2ralog(S1/r)  at  every  point  of  the  cylindrical 
surface,  we  shall  see  that  the  potential  function  due  to  the 
two  distributions  has  at  any  point  Q  outside  the  cylinder 
the  value  2m'  log  (CQ)  =  2  (M  +  m)  log  CQ  and  at  any  point 
within  the  value 

2  m'  log  r  +  2  m  log  8X  /  r  —  2m  log  (r^  /  r2) 

=  2  (M  +  w)  log  r  +  2  m  log  B^/rr^ 

On  the  cylindrical  surface  the  potential  function  has  the 
constant  value  2  (Jf  -f-  m)  log  ^  and  the  surface  density  at  any 
part  of  it  is,  as  before,  (M+  m)  /2  TIT  —  m  (8X2  —  r2)/2  Tmy2,  or 
(^+ra)/27rr  + w(822-r2)/27r?T22.  What  is  the  physical 
meaning  of  the  special  case  where  M  +  m  =  0  ? 

205.  Let  <£  (a?,  ?/)  be  a  logarithmic  potential  function  due 
to  a  body  distribution  of  density  p  through  an  infinite 
cylinder  the  right  section  of  which  made  by  the  xy  plane 
is  the  region  T,  together  with  a  superficial  distribution  of 
density  o-  on  an  infinite  cylindrical  surface  the  right  section 
of  which  is  the  curve  s  in  the  xy  plane.  Let  a=/i  (x,  y) 
and  /3=/2  (x,  y)  be  any  two  conjugate  functions  analytic  in 
the  region  considered,  and  form  arbitrarily  the  new  function 
3>  (x,  y)  =  4>  [fi(x,  y),f2(x,  y)~\  by  substituting  for  x  and  y  in  <£, 
a  and  ft  respectively.  To  avoid  confusion  call  the  rectangular 
Cartesian  coordinates  in  thfe  plane  in  which  T  and  s  are 
drawn  a  and  ft,  instead  of  x  and  y,  and  draw  a  new  xy  plane 
in  which  to  study  the  new  function  <l>.  In  this  second  figure 
the  curves  in  which  a.=fi(x,y),  P=fz(x,y)  are  constant 
form  a  set  of  orthogonal  curvilinear  coordinates.  A  point 
P  which  in  the  first  diagram  has  Cartesian  coordinates 
(oo,  A,)  is  said  to  be  transformed  into  a  point  P'  in  the  new 
diagram,  the  curvilinear  coordinates  of  which  are  (OQ,  /?0).  The 
Cartesian  coordinates  of  P'  are  (aj0,  y0)j  where  /i  (x0,  y0~)  =  a0) 
/2  (xo>  2/o)  =  Po-  It  is  evident  that  4>  (x,  y)  has  the  same 


MISCELLANEOUS    PROBLEMS.  407 

numerical  value  at  P'  that  <£(a,  ft)  has  at  P.  The  points 
which  lie  on  the  curve  s  in  the  old  diagram  are  transformed 
into  points  which  lie  on  a  curve  s'  in  the  new  diagram,  so  that 
the  curve  s  is  transformed  into  the  curve  s'  and,  similarly,  the 
region  T  into  the  region  T'.  It  is  evident  from  the  proper 
ties  of  conjugate  functions  that  two  curves  which  cut  at  an 
angle  6  at  a  point  P  in  the  old  diagram  transform  into  two 
curves  which  cut  each  other,  in  general,  at  the  same  angle 
at  the  point  P'.  Show  that  3>  is  the  logarithmic  potential 
function  due  to  a  body  distribution  through  the  infinite 
cylinder  of  which  T'  is  the  cross-section,  together  with  a  sur 
face  distribution  on  the  cylindrical  surface  of  which  s'  is  the 
trace.  Show  also  that  if  k2  represents  either  of  the  two  equal 
quantities  (Dxa)2  +  (Dva)2,  (D^y  +  (Dyft)*,  the  numerical 
relations,  at  corresponding  points  in  the  two  diagrams,  of  the 
corresponding  elements  of  arc  and  area,  of  the  corresponding 
values  of  the  volume  and  surface  density,  etc.,  etc.,  are  truly 
given  by  the  equations 

ds  =  /ids'-,  dA  =  h*dA';  ph?  =  p'  •  ah  =  o-'  ; 

<£  =  <£;  A<£  =  A<£  ;  h  D&  =  ZK<£,  7r  A2<£  =  A23>  ; 

h  Dn$  =  Da&  ;  p  dA  =  p'dA'  ;  a-  ds  =  cr'ds'. 

206.  Given  in  a  plane  two  circles  of  radii  a  and  b  respec 
tively,  which  have  no  points  in  p  omnion,  it  is  possible  to  find 
two  points  (ft,  Q2)  on  the  line  which  joins  their  centres 
(A,  fi),  such  that  if  ^  and  r.2  represent  the  distances  from 
ft  and  ft  of  any  moving  point,  both  circles  belong  to  the 
family  of  curves  represented  by  the  equation  ?-1/;-2  =  c.  Show 
that  if  AB  =  d,  and  if  the  circles  are  mutually  exclusive,  ft 
and  ft  are  between  A  and  B,  and 


AQl=(a2  +  d*  -b*-  K)/2d,  BQ,  =  (#»  +  d>  ->* 

where  R*  =  (a2  -  b*  -  d*y  -  4  b*d*.     If  one  of  the  circles  lies 
within  the  other  and  if  a  >  b,  ft  and  ft  lie  beyond  B  on  the 


408  MISCELLANEOUS    PROBLEMS. 

line  ABj  Ql  is  within  both  circles,  and  Q2  outside  both.  In 
this  case, 

AQl  =  (a*  -b2  +  d2-  R')/2  d,  BQ,=  (a2  -b2-d2  +  K)  /2  dt 

where  R'2  =  (b2  —  a2  —  d2)2  —  4  a2d2.  In  the  second  case  the 
four  points  of  contact  of  tangents  drawn  from  Q2  to  the  two 
circles  lie  on  a  straight  line  through  Ql.  What  is  the  corre 
sponding  fact  in  the  case  first  treated  ?  Consider  the  special 
case  where  a  =  b  =  i-  d. 

Prove  that  the  values  of  AQl}  BQ2  given  in  the  subjoined 
table  are  correct  and  draw  to  scale  a  diagram  for  each  of  the 
four  examples. 


a 

b 

d 

AQ, 

BQ* 

1 

2 

4 

0.35 

1.09 

4 

2 

1 

1.37 

10.62 

3 

1 

1 

1.14 

6.85 

5 

3 

1 

1.62 

14.38 

Given  a  circle  of  radius  a,  with  centre  at  A,  and  a  straight 
line  in  its  plane  at  a  distance  d  from  A  greater  than  a  ;  the 
line  and  the  circumference  belong  to  the  family  of  curves 
rl/r2  =  c,  where  rt  and  rz  represent  the  distances  from  two 
points  (Qlt  $2)  equidistant  from  the  line  011  opposite  sides  of 
it  and  lying  on  the  perpendicular  to  the  line  drawn  through  A. 
Show  that  if  Q1Q2  =  2  m,  m2  ^  d2  -  a2. 

207.  If  r2  =  (x  +  a)2  +  y\  r2  =  (x-  a)2  +  y\  tan  0,  =  y/(x  +  a) , 
tan  02  =  y  I  (x  —  a)  ;  (j>  =  A  log^/rg),  and  \l/  =  A(Ol  —  02)  are 
conjugate  functions,  and  if,  moreover, 

„ .  ,  ,  a(c2  4-  1)  2  a  2  ac 

c2  =  e2^^a=     ^2_i     ;    a-^-ni'    andr=±^— -, 

where  the  upper  or  lower  sign  is  to  be  used  according  as  c2  is 
greater  or  less  than  unity  ;  (x  —  a)2  -f  (y  —  O)2  =  ?-2,  and  the 
curves  of  constant  <£  are  the  circles  surrounding  the  points 
(a,  0)  and  (—  a,  0)  represented  in  Fig.  59.  Values  of  c 


MISCELLANEOUS    PROBLEMS.  409 

between  0  and  1  correspond  to  circles  which  lie  to  the  left 
of  the  y  axis,  and  positive  values  of  c  greater  than  1  to  circles 
on  the  right  of  the  y  axis. 

When  c  >  1,  c  =  (a  +  a)/r,  and  a  =  + 

but  when      c  <  1,  c  =  —  (a  +  a)  /r,  a  =  -f- 

On  two  circumferences  of  the  system,  of  equal  radii,  on  oppo 
site  sides  of  the  y  axis,  rx/r2  has  reciprocal  values. 

Using  these  formulas,  prove  that  the  charge  per  unit  length 
on  a  long  cylindrical  wire  of  radius  0.5  centimetre,  kept  at 
potential  unity  at  an  axial  distance  of  a  =  600  centimetres 
from  an  infinite  plane  kept  at  potential  zero,  is  0.06424  units. 
In  this  case  c  is  about  2400,  a  about  599.9998,  and  A,  0.128. 
Show  also  that  if  r  =  0.5  and  a  =  10 ;  a  =  9.988,  c  =  39.975, 
A  =  0.271,  but  that  if  r  =  0.5  and  a  =  1 ;  a  =  0.866,  c  =  3.732, 
A  =  0.759.  It  is  to  be  noticed  that  300  volts  are  equivalent 
to  1  electrostatic  unit  of  potential  difference,  1  microfarad  to 
900,000  electrostatic  units  of  capacity,  1  ohm  to  I/ (9  X  1011) 
electrostatic  units  of  resistance,  1  ampere  and  1  coulomb  to 
3  X  109  corresponding  electrostatic  units. 

In  general,  if  an  infinite  conducting  cylinder  of  revolution 
kept  at  potential  F0  be  placed  with  its  axis  parallel  to  an 
infinite  conducting  plane  at  a  distance  a  from  it,  the  charge 

per  unit  of  length  is  ^  F"0/log^-Jl — L?  and  the   surface 

density  is  inversely  proportional  to  the  distance  from  the  plane. 
208.  A  condenser  is  formed  by  two  long  conducting  circular 
cylinders,  one  of  which  is  entirely  inside  the  other.  Prove 
that  if  r  and  r'  are  the  radii,  d  the  distance  between  the 
axes,  and  2  a  the  distance  between  the  limiting  points  of 
the  coaxial  system  to  which  the  cylinders  belong,  the  inverse 
of  the  capacity  per  unit  length  is 


MISCELLANEOUS    PROBLEMS. 

209.  Electricity  is  distributed  in  equilibrium  over  the 
surface  of  an  infinitely  long  right  cylinder,  the  cross-section 
of  which  is  x*  +  y*  =  a4.  Prove  that  the  attraction  at  any 
external  point  (r,  ff)  is  inversely  proportional  to 


and  that  its  direction  makes  with  the  axis  of  x  an  angle 

[Clare  College.] 


210.  A  long,  right  circular  cylinder  of  radius  a  is  placed 
with  axis  parallel  to  a  plane  at  potential  zero.     Show  that 
the  mutual  attraction  per  unit  length  of  the  cvlinder  between 
it  and  the  plane  is  E*/  Vr2  —  a1,  where  c  is  the  distance  of 
the  axis  of  the  cylinder  from  the  plane  and  E  the  quantity 
of  electricity  on  the  unit  length  of  the  cylinder.     [M.  T.] 

211.  Av  Q,  A*  three  points  in  order  on  a  straight  line,  such 
that  J,£  =  m,  QA^  =  n,  have  charge* 

*!  =  A  Vm(m  +  n),  e^  =  —  X  V»™,  «.  =  X  Vn  (m  -h  n) 

respectively.  The  charges  e^  %  produce  potential  zero  on  a 
spherical  surface  ^  of  radius  a  =  Vwi  (i»  +  n)  with  centre  at 
Av  and  the  charges  e^  e^  produce  potential  zero  on  a  spherical 
surface  Sj  of  radius  b  =  Vr<  ^m  -f-  n)  with  centre  at  A*  Sl  and 
5,  cut  each  other  orthogonally  and  together  form  the  equi- 
potential  surface  X  due  to  e<»  ev  and  e*  Show,  by  a  method 
analogous  to  that  of  Section  65.  that  the  resultant  force  at  any 
point  P  of  SH  due  to  e^  and  «*  is  directed  towards  Al  and  is 
numerically  equal  to  Xfr3/  a  -  A^P*,  so  that  the  whole  force  at 
P  has  the  direction  A^  and  the  intensit 


Fl  =  X[l/a  -  tf/a 

Find  a  similar  expression  for  the  whole  force  at  anv  point  of 
the  surface  £,.  Show  by  the  help  of  Section  31  that  the  surface 
taken  over  the  larger  segment  of  S^.  of  the  normal 


MISCELLANEOUS    PROBLEMS.  411 

components  of  the  forces  due  to  ev  e0,  and  ez  respectively  are 
2ire1(l  +  m/a),  2vew  2^(1  -  n/b).  Prove  that  if  ew  el} 
and  ez  were  distributed  on  the  surface  composed  of  the  larger 
segments  of  S1  and  S2  according  to  the  law  a  =  F/4=  IT,  the 
surface  woulgL  be  at  potential  A,  and  there  would  be  no  density 
at  the  circle  of  intersection  of  Sj_  and  &.  The  charge  under 
these  circumstances  on  the  larger  segment  of  Sl  would  be 

\  1X1  +  m/a)+e0  +  e,(l  -  »/&)], 
or  \ X (a  -f-  &  4-  m  —  Vwm  —  ri), 

or  £A6[1  +  8  +(8S  -  S  -  I)/  Vl  -f  £-'], 

where  B  =  a/b.  If  &  is  very  large  compared  with  a,  the  larger 
segment  of  Sl  becomes  nearly  hemispherical ;  its  charge  is 
about  3  Acr /4  6  and  its  mean  density  3\/Sirb.  The  mean 
density  on  St  when  the  ratio  of  a  to  b  is  small  is  nearly  equal 
to  A.  (4  br  —  3  a-}  /1 6  Trbs.  If  a/b  =  0,  we  have  a  hemispherical 
boss  on  an  infinite  plane  ;  the  ratio  of  the  average  densities  of 
the  charges  on  the  boss  and  the  plane  is  3/2. 

212.  A  point  charge  e  at  (4  b,  0,  0)  and  a  point  charge  —  e 
at  (—  4  b,  0,  0)  keep  the  plane  x  =  0  at  potential  zero.  Show 
that  if  the  system  be  inverted,  using  the  point  (—  2  b,  0,  0)  as 
centre  of  inversion  and  2  b  for  radius  of  inversion,  we  obtain 
a  spherical  surface  of  radius  b,  with  centre  at  (—  b,  0,  0),  kept 
at  potential  zero  by  the  charge  —  e  at  the  point  (—  46,  0,  0), 
and  the  charge  -J  e  at  (—  |  b,  0,  0) :  this  is  the  problem  of 
Section  65.  If  the  centre  of  inversion  were  (—  4  b,  0,  0)  and 
if  a  were  4  b,  we  should  obtain  by  inversion  a  spherical  sur 
face  of  radius  2  b,  with  centre  at  (—  2  b,  0,  0)  at  potential  zero, 
under  a  charge  1  e  at  its  centre,  and  an  infinite  charge  at  infinity 
which  lowers  the  potential  function  at  all  finite  points  by  e/4  b. 
If  this  last  were  omitted,  the  value  of  the  potential  function  on 
the  spherical  surface  would  be  e/4  b,  as  is  otherwise  evident. 
Invert  a  spherical  surface  uniformly  charged  with  density  o-, 
using  any  point  not  its  centre  as  centre  of  inversion. 


412  MISCELLANEOUS    PROBLEMS. 

213.    If  z  =  w  -f-  ew  ;  x  =  <f>  +  6$  cos  \J/j  y  =  ^  +  e^  sin  «^,  the 
slopes  of  curves  of  constant  <£  are  given  by  the  equation 

dy  I  'dx  =  —  (1  +  e*  cos  \j/)  /e*  sin  ^, 
and  the  slopes  of  curves  of  constant  ^  by  the  equation 
dy  /dx  =  e^  sin  ^/(l  +  e*  cos  i/^). 


Show  that  :  (1)  The  curve  $  =  0  is  the  axis  of  a:,  and  on  it 
x  =  </>  -f  e$  ;  large  positive  values  of  x  and  <f>  correspond,  and 
large  negative  values  correspond.  (2)  If  if/  =  ?r,  y  =  TT  and 
x  =  (f>  —  e^  :  the  maximum  value  of  x  is  —  1,  so  that  the 
curve  \j/  =  TT  is  so  much  of  the  line  y  =  TT  as  lies  to  the  left 
of  x  =  —  1.  (3)  The  curve  if/  =  —  TT  is  so  much  of  the  line 
y  =  —  TT  as  lies  to  the  left  of  x  =  —  1.  (4)  The  curve  </>  =  0 
has  the  slope  —  ctn  1  \j/  and  passes  through  the  points  (—  1,  TT), 
(-  1,  -  TT),  (1,  0),  (0,  i  TT  +  1),  (^  J  IT  +  i  V3).  (5)  The  curve 
<£  =  —  m,  where  m  is  any  positive  quantity,  lies  to  the  left 
of  <£  =  0,  between  the  lines  y  =  —  TT,  y  =  +  TT:  it  has  the  slope 
—  (em  +  cos  \j/)/sin  \l/,  which  is  infinite  when  y  =  Oori/=  —  ?ror 
y  =  TT,  and  has  the  minimum  value  Ve2m  —  1,  so  that  for  values 
of  w  greater  than  3  the  line  is  hardly  distinguishable  from  a 
straight  line  parallel  to  the  axis  of  y,  at  a  distance  of  (e~m  —  m) 
to  the  left.  (6)  The  curve  <£  =  +  m,  where  m  is  a  positive 
quantity,  lies  to  the  right  of  the  curve  </>  —  0,  it  cuts  the  axis 
of  y  perpendicularly  and  meets  (but  does  not  cross)  the  lines 
y  =  —  ir,  y  =  TT  from  without.  The  curves  for  which  <£  has 
the  values  1,  2,  3,  4,  5,  6,  7  meet  y  =  TT  at  points  the  abscis 
sas  of  which  are  -  1.72,  -  5.39,  -  17.08,  -  50.60,  -  143.4, 
-  397.5,  -  1090  respectively.  (7)  If  so  much  of  the  planes 
y  =  4-  TT,  y  —  —  TT  as  lie  to  the  left  of  x  =  —  1  be  considered 
conducting  and  be  charged  to  potential  +  TT  and  —  TT  respec 
tively,  \j/  represents  the  potential  function  in  the  air  near 
them.  In  this  case  the  charge  on  either  side  of  a  strip  of 
either  plane  between  the  planes  x  =  —  xl}  x  =  —  x2  is  equal, 
per  unit  length  of  the  strip  parallel  to  the  z  axis,  to  the 


MISCELLANEOUS    PROBLEMS.  413 

difference  on  that  side  of  the  plane  between  the  values  of  <£ 
for  x  =  —  xl  and  x  =  —  xz,  divided  by  4  ?r.  On  a  strip  of  the 
plane  y  =  TT,  between  x  =  —  1  and  x  =  —  50.6,  there  is,  per 
unit  height  of  the  strip,  a  charge  I/TT  on  the  upper  side 
and  of  50.6  /  4  TT  on  the  under  side:  the  charge  on  correspond 
ing  portions  of  y  =  —  TT  being  equal  and  opposite  to  these. 
[Helmholtz,  CrellJs  Journal,  Vol.  LXX.] 

State  carefully  some  problem  in  electrostatics  which  might 
be  solved  by  the  use  of  the  function  z  =  A  [cw  +  e010]. 

A  condenser  consists  of  two  very  thin,  large,  plane,  metal 
sheets  of  the  same  area  parallel  to  each  other  at  a  distance 
of  1  millimetre.  The  dielectric  is  air  and  the  difference  of 
potential  between  the  plates  is  1  electrostatic  unit  (300  volts). 
Show  that  the  density  of  the  charge  2  millimetres  from  the 
edge  is  about  5/2  TT  per  square  centimetre  on  the  inside  of 
the  plate. 

Discuss  at  length  the  function 


sin  (mr) 

where  n  is  any  real  constant  between  0  and  £  [Harris,  Ann. 
of  Math.,  1901],  and  state  some  problems  of  electrostatics 
which  can  be  solved  by  its  aid. 

214.  Three  closed  surfaces  1,  2,  3  in  order  are  equipotential 
surfaces  of  an  electrostatic  field  in  air.     If  an  air  condenser 
were  constructed  with  the  faces  1,  2,  its  capacity  would  be  A,  but 
if  the  faces  were  2,  3,  its  capacity  would  be  B.     Show  that  if 
a  condenser  were  constructed  with  faces  1,  3  while  a  homoge 
neous  dielectric  of  inductivity  /x  filled  the  space  1,  2,  and  a 
second  dielectric  of  inductivity  /x'  the  space  2,  3,  the  capacity 
of  this  condenser  would  be  (7,  where  1/C=  1/pA  -f  1  /  '  ^B. 

215.  A  condenser  is  formed  of  two  concentric  spherical  con 
ducting  surfaces  of  radii  a  and  c,  separated  by  two  dielectric 
shells  bounded  by  a  spherical  surface  of  radius  b  concentric 
with  the  conducting  surfaces.     Prove  that   if   in   one  shell 


414  MISCELLANEOUS    PROBLEMS. 

^  =  m/r2  and  in  the  other  w'/r2,  the  capacity  of  the  condenser 
is  mmf/[m'(£  —  a)  +  m(c  —  £)]. 

216.  If  the  space  between  two  closed  equipotential  surfaces 
in  air  be  filled  with  a  dielectric  the  inductivity  of  which  is 
either  uniform  or  a  scalar  point  function  the  level  surfaces  of 
which  coincide  with  the  equipotential  surfaces  of  the  field,  the 
potential  function  without  the  shell  will  be  unchanged,  but  its 
value  within  will  be  increased  by  a  constant. 

217.  An  infinite  dielectric  is  bounded  by  an  infinite  con 
ducting  plane  which  is  maintained  at  a  potential  Ar2,  where  r 
is  the  distance  from  a  point  0  in  the  plane.     Prove  that  if  the 
inductivity  of  the  dielectric  varies  as  the  distance  z  from 
the  plane,  the  potential  at  any  point  is  X  (u2  —  z2),  where  u 
is  the  distance  from  an  axis  drawn  through  0  perpendicular 
to  the  conducting  plane. 

218.  A  distribution  of  matter  M  consists  of  two  portions  M1} 
in  a  homogeneous   medium  of  inductivity  /x1?  and  M2,  in  a 
homogeneous  medium  of  inductivity  /x2  surrounding  the  other 
medium  and  reaching  to  infinity.     An  equipotential  closed 
surface   Si  surrounds  MI,  excludes  M2,  and   lies  wholly  in 
the  first  medium,  a  second  closed  equipotential  surface   S2 
surrounds  MI,  excludes  M2,  and   lies  wholly  in  the  second 
medium.     Prove  that  if  r  is  the  distance  from  a  fixed  point  0, 
if  normals  are  drawn  outward  on  $2  and  inward  on  Si,  and  if 
dri  and  drz  are  elements  of  space  within  Si  and  without  $2 
respectively, 


and        4  TT^  V0  =  -  t*2dS2  +  4 
if  0  is  without  S2,  and 


MISCELLANEOUS    PROBLEMS.  415 


and       4  TT/X,  (  V0  -  rSl)  =  - 
if  0  is  within  Si. 

Show  from  these  equations  that  if  S,  the  surface  of  separa 

tion  of  the  two  media,  is  equipotential,  J  J  J  ^p"  is  equal 

to  /x2  V0  if  0  is  without  S,  and  to  ^  V0  +  (p.2  —  ^  Vs  if  0  is 
within  £  Give  physical  interpretations  to  these  last  results. 
How  is  the  force  at  any  outside  point  affected  by  the  sub 
stitution  of  one  homogeneous  dielectric  for  another  in  the 
whole  region  bounded  by  S? 

219.  The  open  surface  S  is  a  surface  of  zero  potential  due  to 
a  distribution  J/x  in  an  infinite  homogeneous  medium  of  induc- 
tivity  fj.i  on  the  right  of  S,  and  to  a  distribution  M2  in  an 
infinite  homogeneous  medium  of  inductivity  /x.2  on  the  left  of 
S.     S  is  the  common  boundary  of  the  two  media.     Show  that 

if  r  is  the  distance  from  a  fixed  point  0,  j  J  J  ^—^-  =  ^  V  or 

/x2  Vj  according  as  0  is  to  the  right  or  to  the  left  of  S. 

220.  The  function  W  so  vanishes  at  infinity  that  i&DrW, 
where  r  is  the  distance  from  any  finite  point,  is  not  infinite. 
The  normal  derivative  of   W  is  given  at  every  point  of  an 
infinite  plane.    Prove  that  if  W  is  harmonic  everywhere  in  the 
space  on  one  side  of  the  plane,  it  is  determined  in  that  region. 
Prove  also  that  if  W  is  harmonic  in  the  region  on  one  side 
of  the  plane  except  at  the  given  points  -Pj,  P2,  P3,  •  •  •  ,  PnJ  at 
each  of  which  it  becomes  infinite  in  such  a  manner  that,  if  rk  is 
the  distance  from  Pk,  and  if  mk  is  a  constant  belonging  to  this 

point,  W  —  —  -  is  harmonic  at  Pk,   W  is  determined  in  the 

region  in  question. 

221.  Two  homogeneous  media  of  inductivities  /^  and  ^  have 
a  plane  surface  of  separation  but  are  otherwise  unbounded. 
In  the  first  medium  at  a  point  P  at  a  distance  a  from  the 


416  MISCELLANEOUS    PROBLEMS. 

common  surface  S  of  the  two  media  is  a  charge  e  =  ^e. 
At  Q,  any  point  on  S,  the  force  due  to  this  charge  has  the 
normal  component  ea/(PQ)s,  or  8,  pointing  into  the  second 
medium.  If  jVj  and  N2  are  the  normal  components  of  the 
whole  force  at  Q  pointing  into  the  two  media,  and  if  o-'  is 
the  apparent  density  of  the  surface  charge  on  the  plane 

at  ft 

N,  =  2  7r<r'  -  8,  Nz  =  2  TTO-'  +  3, 

and 


whence  N,  =  S  [(^  -  /x2)  /  (^  +  ^  -  1], 

and  ^  =  2^8/0^  +  ,*,). 

Prove  that  ^  might  be  caused  by  an  apparent  charge 
(/A!  —  /x2)  e  /  (/A!  +  /u,2)  at  P',  the  image  of  JP  in  the  plane, 
together  with  an  apparent  charge  e  at  P  and  that  N2  might 
be  due  to  an  apparent  charge  2  /xxe  /  (^  4-  /u,2)  at  P.  Hence 
show  by  the  aid  of  the  theorem  stated  in  the  last  problem 
that  the  potential  functions  due  to  these  apparent  charges  are 
identical  (one  in  the  first  medium,  the  other  in  the  second) 
with  the  values  of  the  actual  potential  function  in  the  case 
described  in  this  problem.  The  charge  at  P  is  urged  towards 

the  dielectric  with  the  force  -~  -  —  -  —  • 

*«    ^2  +  ^ 

222.  Using  the  notation  of  Section  62,  let  the  plate  A  of  a 
spherical  condenser  be  charged  with  m  units  of  positive  elec 
tricity  and  separated  from  the  plate  J5,  which  is  put  to  earth, 
by  a  spherical  shell  of  radii  r  and  rf  made  up  of  a  given 
dielectric.  Let  us  first  ask  ourselves  what  the  effect  of  the 
dielectric  would  be  if  it  consisted  of  extremely  thin  concentric 
conducting  spherical  shells  separated  by  extremely  thin  insu 
lating  spaces.  It  is  evident  that  in  this  case  we  should  have 
a  quantity  —  m  of  electricity  induced  on  the  inside  of  the 
innermost  shell,  a  quantity  +  m  on  the  outside  of  this  shell, 
a  quantity  —  m  on  the  inner  surface  of  the  next  shell,  a 


MISCELLANEOUS    PROBLEMS.  417 

quantity  +  m  on  the  outside  of  this  shell,  and  so  on.  If 
there  were  n  such  shells  in  the  dielectric  layer,  and  n  +  1 
spaces,  and  if  S  were  the  distance  from  the  inner  surface  of 
one  shell  to  the  inner  surface  of  the  next,  and  AS  the  thick 
ness  of  each  shell,  the  value,  at  the  centre  of  J,  of  the  poten 
tial  function  due  to  the  charges  on  these  shells  would  be 

V'=ml      "  *  l 


8       r  -  AS  +  8       r  +  2  S       /•  -  AS  +  2  S 
+  •••  +       1    . 


r  +  -wS       r  -  AS  +  ?*S  J 
(H-28)(r- A8+28)  +  '  "  J 


This  quantity  lies  between 


r*r 


but  these  differ  from  each  other  by  less  than  c  = 

fi—u—v&dx  '  '« 

so  that  —  m\j  — ,  which  is  easily  seen  to  lie  between 

G  and  H,  differs  from  VA'  by  less  than  e.     If,  then,  S  is  very 

small  in  comparison  with  rand  ?•.,  V ,'  differs  from  m\( j 

V<      V 

by  an  exceedingly  small  fraction  of  its  own  value. 

This  shows  that  the  effect,  at  the  centre  of  A,  of  such  a 
system  of  conducting  shells  as  we  have  imagined  would  be 
practically  the  same  as  if  a  charge  —  ?n\  were  given  to  the 
inner  surface  of  the  dielectric,  and  a  charge  4-  m\  to  its  outer 
surface,  while  the  charges  on  the  surfaces  of  the  thin  shells 
within  the  mass  of  the  dielectric  were  taken  away.  That  is, 
the  value  of  the  potential  function  in  A  would  be 

m(l  —  \)(-      -  J  instead  of  m  (-      -Y 


418  MISCELLANEOUS    PROBLEMS. 

Such,  a  system  of  shells  introduced  into  what  we  have  hitherto 
supposed  to  be  the  electrically  inert  insulating  matter  between 
the  two  parts  of  a  spherical  condenser  would  increase  the 
capacity  of  the  condenser  in  the  ratio  of  1  to  1  —  A.  It 
is  to  be  noticed  that  X  is  a  proper  fraction  :  X  =  0  and  X  =  1 
would  correspond  respectively  to  a  perfect  insulator  and  to  a 
perfect  conductor. 

If  the  coatings  of  a  parallel  plate  air  condenser  be  in  the 
planes  x  =  0,  x  =  a,  and  if  the  first  have  a  uniform  superficial 
charge  of  density  —  <r  and  be  kept  at  potential  zero,  the 
potential  function  in.  the  air  between  the  plates  is  evidently 
47TO-X.  Show  that  if  a  number  of  plane  plates  of  metal  of 
small  thickness  AS  be  uniformly  distributed  between  the  coat 
ings  parallel  to  the  yz  plane  so  as  to  be  separated  from  each 
other  by  air  spaces  of  thickness  (1  —  A)  8,  the  capacity  of  the 
condenser  will  be  increased  in  the  ratio  of  //,  to  1,  where 
/JL  =  1/(1  —  A).  Show  also  that  if  8  be  made  infinitesimal 
and  A  a  function  of  x,  we  have  between  the  coatings  in  the 
limit,  pDx  V  =  4  TTO-,  or  Dx  (^Dx  F)  =  0,  the  differential  equa 
tion  which  F  would  satisfy  in  a  real  dielectric  of  inductivity 
varying  with  x.  Treat  again,  on  the  assumption  that  A 
varies  with  r,  the  case  of  the  spherical  condenser  considered 
above. 

223.  The  potential  function  F  due  to  an  electrical  or  mag 
netic  distribution  in  an  inductive  medium,  may  be  computed 
according  to  the  Newtonian  Law  by  taking  into  account  both 
the  intrinsic  and  the  induced  charges.  If  p0  and  <r0  are  the 
intrinsic  volume  and  surface  densities,  and  if  the  integrations 
extend  all  over  the  space  where  p0  and  o-0  are  different  from 
zero,  the  potential  energy  of  the  distribution  is  usually  written 

4 


or 


MISCELLANEOUS    PROBLEMS.  419 

Why  should  not  the  apparent  volume  and  surface  densities 
be  used  in  finding  the  energy  by  the  equation 

E  = 

Answer  this  question  fully,  using  an  illustrative  numerical 
example  to  explain  your  assertions. 

Assuming  that  the  energy  of  an  electrostatic  field  would  be 
mathematically  accounted  for  011  the  supposition  that  every 
volume  element  of  space  at  which  the  intensity  of  the  field 
is  F  contributes  F2/8  TT  times  its  volume  to  the  whole  amount, 
show  that  if  a  tube  of  force  be  cut  into  cells  by  a  set  of  equi- 
potential  surfaces  drawn  at  equal  potential  intervals,  these 
cells  contain  equal  amounts  of  energy.  Show  how  to  divide 
all  space  up  into  unit  energy  cells.  Discuss  the  mechanical 
action  on  a  charged  conductor  in  an  electric  field  on  the 
assumption  that  there  is  tension  along  the  Faraday  tubes 
which  abut  on  the  conductor,  such  that  the  normal  pull  on 
the  conductor  per  square  centimetre  of  its  surface  is  F*/8  TT. 
Discuss  the  pressure  at  right  angles  to  the  Faraday  tubes  in  a 
dielectric. 

224.  The  space  between  two  concentric  spherical  surfaces, 
the  radii  of  which  are  a  and  b  and  which  are  kept  at  potentials 
A  and  B,  is  filled  with  a  heterogeneous  dielectric,  the  induc- 
tivity  of  which  varies  as  the  nth  power  of  the  distance  from 
their  common  centre.  Show  that  the  potential  function  at 
any  point  between  the  surfaces  is 

(Aan+l  -  Bbn+l)/(an+l  -  bn+l)  -  a"+lb"+l  (A  -  B)  /  r"+1  (an+l  - 


225.  A  condenser  is  formed  of  two  concentric  spherical  con 
ducting  surfaces  separated  by  a  dielectric.  This  dielectric 
consists  of  three  shells  bounded  by  spherical  surfaces  of  radii 
TI,  r2,  r3,  and  r4,  concentric  with  the  conductors.  The  induc- 
tivities  of  the  inner  and  outer  shells  are  equal  to  /n1?  and  that 


420  ,       MISCELLANEOUS    PROBLEMS. 

of  the  intermediate  shell  is  /x2.     Show  that  if  C  is  the  capacity 
of  the  condenser, 


ra 

226.  The  plates  of  a  condenser  are  two  confocal  prolate 
spheroids  and  the  inductivity  of  the  dielectric  is  A  j  p,  where 
p  is  the  distance  of  any  point  from  the  axis.     Prove  that  the 
capacity  of  the  condenser  is 

IT  A  I  [log  (a,  +  bj  -  log  («  +  &)], 

where  a,  b  and  alt   ^  are   the  semiaxes   of  the   generating 
ellipses. 

227.  The  plates  of  a  condenser  are  the  closed  metallic  sur 
faces   Sl  and  S2.     When   ^  is  at  potential  zero  and  S2  at 
potential  F2,  the  potential  function  in  the  air  between  them 
is  given  by  the  equation   V  =  f(x,y,z)*     The  tube  of  force 
based  on  a  portion  ($/)  of  ^  abuts  on  a  portion  ($2')  of  S2. 
If  the  air  in  this  tube  were  displaced  by  a  homogeneous  dielec 
tric  of  inductivity  /A,  and  if  the  charges  on  £/  and  S2'  were 
increased  in  the  ratio  /*,,  while  the  charges  on  the  remainder 
of  Sl  and  $2  were  unchanged,  would  the  force  at  every  point 
be  unchanged  ?     Would  there  be  a  discontinuity  in  the  sur 
face  density  of  the  apparent  charge  on  $i  at  the  boundary 
of  S^? 

228.  How  many  square  centimetres  of  tin  foil  must  be  used 
in  making  a  single  parallel  plate  condenser  of  one  microfarad 
capacity,  if  the  two  sheets  of  foil  are  to  be  separated  from 
each  other  by  paraffined  paper  the  thickness  of  which  is  one- 
fifth  of  a  millimetre,  and  the  specific  inductive  capacity  2  ? 
[72,000  TT.]     Would  the  required  amount  be  the  same  if  the 
condenser  were  made  up  of  a  pile  of  sheets  of  foil  alternating 
with  paper,  the  odd   sheets  forming  one  terminal  and  the 
even  sheets  the  other? 


MISCELLANEOUS    PROBLEMS.  421 

229.    Show  that  the  generalized  Poisson's  Equation, 

Dx(pDxV)  +  D,(jp.DyV)  +  Dz(fi.DzV)  =  -  4,rp, 
is  equivalent  to 


if  £,  77,  £  are  any  orthogonal  curvilinear  coordinates. 

In  the  case  of  spherical  coordinates,  where  hr  =  1,  h0  =  !/;•, 
h^  =  I/ >•  sin  0,  the  equation  is 

sin2  6  •  Dr  (JJL  r-Dr  V)  +  sin  0  •  De  O  sin  ODQ  V) 

+  D+  fa  Dt  V)  =  -  4  7rp>-2  sin2  0, 

and,  in  columnar  coordinates,  where  hr  =  1,  he  =  1/r,  h,  =  1, 

itisr.I>r(^rAn  +  A(^An  +  ^-A(^An  =  -4~7rP^ 

230.  Show  that  if  the  poles  of  a  battery,  made  up  of  a  given 

number  of  equal  cells,  are  to  be  connected  by  a  resistance  R 
greater  than  the  sum  of  the  resistances  of  all  the  cells,  the 
greatest  current  will  traverse  R  when  the  cells  are  joined  up 
in  series  ;  but  that  if  R  is  very  small,  the  cells  should  be  joined 
up  in  multiple  are.  If  R  is  such  that  by  arranging  the  cells 
in  a  certain  number  of  parallel  rows  and  joining  up  the  num 
bers  of  each  row  in  series,  the  resistance  of  the  whole  battery 
can  be  made  equal  to  R,  this  arrangement  will  give  the  maxi 
mum  current. 

231.  A  battery  is  joined  up  in  simple  circuit  with  a  resistance 
R  and  a  galvanometer  of  resistance  G.     After  the  deflection  of 
the  galvanometer  has  been  noted,  an  additional  wire  (or  shunt) 
of  resistance  S  is  placed  across  the  poles  of  the  battery,  and  the 
resistance  R  is  decreased  (to  r)  until  the  galvanometer  deflec 
tion  is  the  same  as  before.     Assuming  that  the  electromotive 


422  MISCELLANEOUS   PROBLEMS. 

force  of  the  battery  remains  constant,  show  that  the  resistance 
of  the  battery  is  — ^ — •     [Thomson.] 

232.  Using  the  potential  function  F  =  c  log  r  +  d,  where  r 
is  the  distance  from  a  fixed  axis,  show  that  the  resistance  of  a 
conductor  bounded  by  two  concentric  circular  cylindrical  sur 
faces  of  radii  a  and  b,  and  by  two  planes,  distant  h  from  each 
other,  perpendicular  to  the  axis  of  the  cylindrical  surfaces,  is 


Apply  the  result  to  the  problem  of  finding  the  resistance  of 
the  liquid  in  a  cylindrical  galvanic  element. 

233.  Using  the  potential  function,  V=  dog(?\/r2),  where 
i\  and  r2  are  the  distances   from  two  parallel    fixed   axes, 
show  how  to  find  (see  Fig.  59  and  Problem  207)  the  resist 
ance  of  a  conductor  bounded  by  two  parallel  planes  and  by 
two  somewhat  eccentric  circular  cylindrical  surfaces  which  cut 
the  planes  orthogonally.     In  the  case  of  an  element  in  which 
the  zinc  electrode  is  a  cylindrical   rod  and  the  copper  elec 
trode  a  cylindrical  shell  surrounding  it,  is  the  resistance  of  the 
liquid  greater  or  less  when  the  zinc  is  eccentric  to  the  copper 
shell  than  when  it  is  concentric  with  it  ? 

234.  If  two  points,  A  and  B,  of  a  network  of  conductors 
which  are  carrying  steady  currents,  be  connected  by  an  extra 
conductor  W,  A  and  B  are  said  to  be  at  the  same  potential  if 
no  current  passes  through   W.     A  is  said  to  be  at  a  higher 
potential  than  B  if  a  current  tends  to  pass  through  W  from 
A  to  B.     In  this  case  the  difference  of  potential  between  A 
and  B  is  defined  to  be  the  electromotive  force  (in  volts)  of  a 
galvanic  cell  which  introduced  into  IF  with  its  positive  pole 
towards  A    would  just   prevent  any   current   from   passing 
through  W. 


MISCELLANEOUS    PROBLEMS.  423 

Three  cells  of  electromotive  force  2  volts,  1  volt,  and  1  volt 
respectively,  and  internal  resistances  of  1  ohm,  2  ohms,  and 
4  ohms  are  joined  up  in  series  with  a  resistance  of  1  ohm. 
Show  that  the  potential  differences  between  the  terminals  of 
the  separate  cells  are  + f ,  0,  and  —  1  respectively.  If  the 
external  resistance  were  9  ohms,  the  corresponding  potential 
differences  would  be  -+-  |,  +  £,  0. 

235.  The  terminals  of    a  compound  condenser  formed  of 
three  simple  condensers,  of  capacity  2  microfarads,  3  micro 
farads,  and  6  microfarads  respectively,  joined  up  in  series, 
touch  the  ends  of  a  linear  conductor  of  22  ohms  resistance 
through  which  a  current  of  3  amperes  is  flowing.     What  are 
the  charges  on  the  single  condensers  ?     Show  that  if  with 
out  loss  of  the  charges  the  condensers  be  disconnected  and 
joined  up  in  parallel  with  their  positively  charged  plates  in 
connection,  the  difference  of   potential  between  the  terminals 
of   the  new  compound   condenser  will  be   18  volts.     What 
charge  will  each  of  the  simple  condensers  have  ?     [66 ;  36,  54, 
108.] 

236.  Prove  that  if  a  condenser   of  capacity  k  farads  be 
charged  to  potential    Q0/k  and  then  discharged  through  a 
large  non-inductive  resistance,  r  ohms,  the  charge  Q  of  the 

condenser  t  seconds  after  the  beginning  of  the  discharge  is 

— t 

QQ  -  e**  ;  and  show  that  not  one  ten-thousandth  part  of  the 
original  charge  remains  after  10 kr  seconds. 

Show  also  that  the  energy  that  has  been  expended  up  to 
the  time  t  in  heating  the  wire  is 

Q<?         -— 

^—(1  —  e  *•'•)  joules. 

237.  The    terminals  of  a  condenser  of  k  farads  capacity 
are  attached  permanently  to  the  poles  of  a  constant  battery 
of  electromotive  force  E  volts  by  leads  of  large  resistance, 
r  ohms.     After  the  condenser  has  become  fully  charged  its 


424  MISCELLANEOUS    PROBLEMS. 

terminals  are  suddenly  connected  together  without  removing 
the  battery  by  a  conductor  of  large  resistance,  R  ohms. 
Assuming  that  the  solution  of  a  differential  equation  of  the 
form  Dty  +  ay  =  b  is  y  =  b/a  +  Se~at}  show  that  at  the 
time  t  the  charge  on  one  of  the  condenser  plates  is 


where  q  =  (R  +  r)  /  rRk. 

238.  A  galvanic  battery  is  composed  of  two  galvanic  cells, 
the  electromotive  forces  of  which  are  el  and  e2  and  the  inter 
nal  resistances  ^  and  62,  joined  up  in  multiple  arc.  The 
poles  of  the  battery  are  connected  by  an  external  resistance 
of  r  ohms.  Show  that  if  C\  and  C2  are  the  strengths  of  the 
currents  flowing  through  the  cells, 


239.  A  galvanometer  of  9  ohms  resistance  is  to  be  furnished 
with  two  shunts,  such  that  when  the  first  alone  is  used  TL 
of  the  current  shall  pass  through  the  instrument,  and  that 
when  both  are  used  in  parallel,  29/30  of  the  current  shall 
pass  through  them.     Prove  that  the  resistance  of  the  second 
shunt  must  be  9/20. 

240.  A  storage  battery  is  used  to  send  a  current  through  a 
cluster  of  incandescent  lamps  arranged  in  multiple  arc.     The 
resistance  of  each  lamp  when  hot  is  100  ohms.     When  10 
lamps  are  used  the  current  through  each  is  1  ampere,  but 
when  20  are  used   this  current  is  only  |J-   of  an  ampere. 
Find  the  resistance  of  the  battery  and  its  connections   and 
show  that  the  electromotive  force  of  the  battery  is  110  volts. 

241.  If  a  number  of  cells  of  different  electromotive  forces 
but  of  equal  internal  resistances  are  joined  up  in  multiple 
arc,  the  battery  thus  formed  is  equivalent,  so  far  as  its  ability 


MISCELLANEOUS    PROBLEMS.  425 

to  send  currents  through  outside  resistances  is  concerned,  to 
a  single  cell  the  electromotive  force  of  which  is  the  mean  of 
the  electromotive  forces  of  the  cells  in  the  battery.  Find  the 
resistance  of  this  equivalent  cell  and  show  that  it  would  be 
more  " effective"  when  doing  a  given  amount  of  external 
work  than  the  battery.  How  much  work  is  done  in  the 
battery  per  second  when  the  external  circuit  is  broken? 

242.  A  certain  uniform  cable  50  kilometres  long  has,  when 
in  good  condition,  a  resistance  of  450  ohms.     The  operator  at 
one  end  finds  that  the  resistance  is  270  ohms  or  350  ohms 
according  as  the  other  end  is  grounded  or  insulated.     Suppos 
ing  the  ground  connections  at  the  two  stations  to  be  good,  so 
that  the  resistance  of  the  earth  is  negligible,  and  assuming 
that  the.re  is  a  single  fault  in  the  cable,  show  that  this  fault 
is  16.67  kilometres  from  the  first  station  and  that  its  resist 
ance  is  200  ohms. 

243.  A  cable  500  kilometres  long  with  stations  A  and  B  at 
its  extremities  has  a  single  fault,  but  is  not  so  much  injured 
that  signals  cannot  be  sent  through  it.     With  cable  insulated 
at  B,  the  operator  at  A  grounds  one  terminal  of  a  large  bat 
tery  and  attaches  the  other  terminal  to  the  cable.     After  this 
has  been  done  the  operators  find  that  the  difference  of  poten 
tial  between  the  cable  and  the  ground  is  200  volts  at  A  and 
40  volts  at  B.     The  cable  at  A  is  then  insulated,  and  one 
terminal  of  a  large  battery  at  B  is  grounded  while  the  other 
is  attached  to  the  cable.     The  difference  of  potential  between 
the  cable  and  the  ground  is  then  300  volts  at  B  and  40  volts 
at  A.     Show  that  the  fault  has  a  resistance  equivalent  to 
that  of  47.62  kilometres  of  cable  and  is  at  190.5  kilometres 
from  A.     Explain  some  way  of  measuring  the  potential  differ 
ences  in  this  case. 

244.  « In  a  network  PA,  PB,  PC,  PD,  AB,  BC,  CD,  DA, 
the  resistances  are 

a,  Ay,  8,  y  +  8,  8  +  a, 


426  MISCELLANEOUS    PROBLEMS. 

respectively.     Show  that  if  AD  contains  a  battery  of  elec 
tromotive  force  Ej  the  current  in  BC  is 

X  (aft  +  78)  E 


2  XV  +  (£8 -ay)2 

where  A  =  a  +  ^  +  y  +  S, 

and  n  =  (3y  +  ya  +  a/8  +  aS  +  /38  +  78." 

245.  Show  that  if  the  edges  of  a  parallelepiped  be  formed 
of  uniform  wire  such  that  the  resistances  of  three  contermi 
nous  edges  are  a,  b,  and  c  respectively,  and  if  a  current  enters 
at  one  angle  and  leaves  at  the  opposite  angle,  the  resistance 
of  the  network  is  £  \_(a  +  b  +  c)  +  abc /  (ab  -f-  be  -f-  ca)]. 

246.  (a)  A  tetrahedral  framework  is  made  of  uniform  wire, 
opposite  edges  being  equal  and  of  lengths  a,  b,  c.     If  a  cur 
rent  enters  and  leaves  the  framework  at  the  ends  of  an  edge 
of  length  a,  the  strengths  of  the  currents  in  the  pairs  of  edges 
of  length  a  are  in  the  ratio 

b(a  +  c)+c(a  +  b'):  b(a+  c)—c(a  +  b). 

[Jesus  College.] 

(b)  Show  that  the  resistance  of  the  whole  framework  is  that 
of  a  length  of  the  wire  equal  to  £[d&/(a  +  0)  +  a0/(a  +  &)]. 

[St.  John's  College.] 

247.  Show  that  if  n  telegraph  poles,  each  of  resistance  Ry 
be  joined  in  pairs,  each  to  all  the  others,  with  wires  of  resist 
ance  r,  and  if  an  electromotive  force  E  be  inserted  in  one  of  the 
wires,  the  current  in  that  wire  is  E{R(n  —  2)  +  r}  fr(nR  +  ?*). 

248.  An  electric  distributing  conductor  6  miles  long  gives 
out  continuously  50  amperes  of  current  per  mile  of  its  length. 
The  end  of  the  conductor  remote  from  the  generator  is  insu 
lated,  while  the  nearer  end  is  kept  at  1000  volts  potential. 
Show  that  if  the   resistance  per  mile  of  the  conductor  is 


MISCELLANEOUS  PROBLEMS.  427 

1  ohm,  the  voltage  at  a  point  on  the  line  x  miles  from  the 
generator  isv  =  5Qx(-  — 6)  +  1000.     Find  the  rate  at  which 

a  given  portion  of  the  line  is  delivering  power. 

249.  A  Wheatstone's  bridge  in  proper  adjustment  consists 
of  four  conductors,  AB,  BC,   CD,  DA,  which  have  respec 
tively  the  resistances  p,  q,  s,  and  r.     The  galvanometer  is 
connected  with  A  and  C  and  the  battery  with  B  and  D.     The 
electromotive  force  of  the  battery  is  E,  and  the  resistance  of 
the  battery  with  its  connecting  wires  is  b.     Prove  that  the 
heat  developed  per  unit  time  in  the  conductor  AB  is  the 

equivalent  of  the  energy  — — —  • 

1    [6  (s +  >•)  +  >•  (? +  s)]2 

250.  A  generator  of  constant  electromotive  force  E  and 
of  constant  internal  resistance  B  is  used  to  charge  a  storage 
battery  which  now  has  an  electromotive  force  e  and  an  inter 
nal  resistance  b.     Show  that  if  the  poles  of  the  storage  bat 
tery  be  connected  by  a  conductor  of  resistance  r,  a  current 

C'  =  (Be  +  bE)  •*•  l(B  +  b)  r  +  Bb] 

will  go  through  this  conductor. 

251.  The  conductors  AB,  BC,  CD,  and  DA  have  the  resist 
ances  p,  q,  r,  and  s  respectively.     A  is  connected  with  C  by 
a  battery  of  internal  resistance  b  and  electromotive  force  e. 
B  is  connected  with  D  by  a  battery  of  internal  resistance  b' 
and  electromotive  force  e'.     Prove  that  if  the  current  in  AC 
is  zero, 

e\b'(p  +  q  +  r  +  s)  +  (p  +  s)(q  +  r)\+e'(pr  -  qs)  =  0. 

252.  A  conductor  of  given  dimensions  made  of  given  material 
has  two  given  portions  Si  and  S2  of  its  surface  kept  at  constant 
potentials  while  the  rest  of  its  surface  is  a  current  surface. 
Show  that  if  V  is  the  potential  function  within  the  conductor, 
when  Si  is  kept  at  potential  Ci  and  S2  at  potential  C2,  and 


428  MISCELLANEOUS    PROBLEMS. 

if   V  is  the  potential  function,  when  S^  is  kept  at  C\   and 
O2  at   t/2  ,  ~  f         „  f  n  \  n         n  n  ' 

_    Ul     ~       U2      rr    ,     °2   ^1  ~    U2^1 

"  r       r  r       r 

L/i  —    ^2  <^i          O2 

253.  One  end  (0)  of  a  straight  wire  of  radius  a  and  length 
I  is  kept  at  potential  F0,  and  the  other  end  ($)  at  potential 
FI-  The  specific  conductivity  of  the  wire  is  K  and  its  resist 
ance  per  unit  length  is  w,  so  that  the  reciprocal  of  w  is  equal 
to  ?ra2K.  The  wire  is  surrounded  by  an  insulating  sheath,  the 
outside  of  which  is  in  contact  with  sea  water  at  potential 
zero.  The  rate  of  leakage  per  unit  length  of  the  wire  or 
cable  through  the  sheath  at  a  place  where  the  potential  of 
the  wire  is  V  is  2  TraX  V.  The  reciprocal  of  2  TraX  is  denoted 
by  W  and  is  called  the  "  insulation  resistance  "  of  the  cable 
per  unit  length.  The  rate  of  flow  of  electricity  into  a  portion 
of  the  cable  of  length  A#,  included  between  two  right  sections, 
the  nearer  of  which  is  distant  x  from  0  and  is  at  potential 
F,  is  —  Kira?Dx  V.  The  rate  of  flow  of  electricity  out  of  this 
element  through  the  sheath  and  from  the  farther  end  is 
-  KTra2  (DXV  +  &XDXV)  4-  2  TraXFA^r.  When  the  current 
is  steady  the  element  neither  gains  nor  loses  electricity 
and  KTTO?&XDX  F  —  2  TraX  FAcc  =  0,  so  that  at  every  point 
DX27  —  (32V=0,  where  (?  =  w/W.  The  general  solution  of 
this  equation  is  of  the  form  F=  AePx  +  Be~Px,  and  if  we 
determine  A  and  B  so  that  F=  F0  when  x  =  0,  and  V=  V±  when 
x  =  I,  we  get  F  =  [  Vl  sinh  (fix)  +  F0  sinh  (3(1-  x)~\  /sinh  (ftl). 

Show  that  if  the  current  which  enters  the  cable  at  0  is  /0 
and  that'which  leaves  it  at  Q  is  Ilf  and  if  /  denote  the  current 
in  the  core  at  a  point  at  a  distance  x  from  0, 


/=  [  F0  cosh  ft  (I  -  x)  -  F!  cosh  (/&)]  /[          Fsinh 
=  70  [  FO  cosh  ft(l-x)-  F!  cosh  0&;)  ]  /  [  r0  cosh  (01)  -  FJ, 

/i  =  [F0  -  F!  cosh  (/?/)]/  [V^TF  sinh  (/?/)] 
=  J0  [  F0  -  F!  cosh  (#)]/[  Fn  cosh  (#)  -  FJ. 


MISCELLANEOUS    PROBLEMS.  429 

Show  also  that  if  the  end  of  the  cable  at  Q  be  insulated 

and  left  to  itself,  VL  =  —  ,  "    .  >  but  if  it  be  put  to  earth, 
cosh  (/?/) 

V=  F0sinh/?(Z-a-)/sinh(0/).      If  in  this  latter  case  the 
cable  were  infinitely  long,  we  should  have  V  =  VQe~^x  and 


The  whole  core  resistance  of  a  certain  cable  1000  miles  long 
is  2000  ohms.  When  one  terminal  of  a  battery  (the  other 
terminal  of  which  is  put  to  earth)  is  attached  to  one  end  of  the 
cable  and  the  other  end  of  the  cable  is  grounded,  the  current  at 
the  sending  end  is  to  the  current  at  the  receiving  end  as  1.1276 
to  1.  Show  that  the  insulation  resistance  of  the  cable  per  mile 
is  8  megohms.  In  the  Atlantic  cable  of  1889,  iv  =  1.54  ohms 
per  kilometre,  and  W=  9,085,000.000  ohms  per  kilometre. 

254.  The  conduction  resistance  of  a  certain  cable  1000  miles 
long  is  10  ohms  per  mile,  whilst  the  insulation  resistance  is 
10  megohms  :  if  the  sending  end  be  at  a  given  potential  and 
the  receiving  end  to  earth,  find  the  whole  charge  of  the  cable 
when  a  steady  current  passes  through  it.     Show  that  if  the 
cable  have  a  leakage  fault  at  the  middle  point  the  resist 
ance  of  which  is  equal  to  that  of  a  length  of  a  miles  of  the 
cable,  the  strength  of  a  steady  current  at  the  receiving  end 

will  be  lowered  in  the  ratio  1  :  1  H  --  •  -  r  •     [~^I-  T.I 

a      ••+•! 

255.  Prove  that  if  any  finite  set  of  algebraic  operations  be 
performed  upon  the  complex  variable  z  =  x  +  yi  taken  as  a 
whole,  and  if  the  result  \io  =f(z)~]  be  written  in  the  form 
<f>(x,  y)  +  i-i(/(x,  ?/),  where  <£  and  \f/,  which  are  said  to  be  con 
jugate  to  each  other,  are  real  functions  of  x  and  y  : 

(a)  Both  <£  and  if/  satisfy  Laplace's  Equation. 

(ft)  Dx$  =  D$  and  Drf  =  -  Dj. 

(c)  At  any  point  P,  the  derivative  of  <£  taken  in  any  direc 
tion  PQ  in  the  plane  xy  is  equal  to  the  derivative  of  \f/  taken 
in  a  direction  PR  at  right  angles  to  PQ,  and  such  that  the 
angle  QPR  corresponds  to  a  counter-clockwise  rotation. 


430  MISCELLANEOUS    PROBLEMS. 

(d)  The  equations  <£(#,  y)  =  c,  \f/(x}  y)  =  c'  represent   two 
families  of  curves  which  cut  each  other  orthogonally. 

256.  Prove  that : 

(a)  If  <f>  and  \j/  are  any  two  conjugate  functions  of  x  and  ?/, 
that  is,  if  <j>  +  i\j/  is  a  function  of  the  complex  variable  x  +  yit 
taken  as  a  whole,  then,  conversely,  x  and  y  are  two  conjugate 
functions  of  <f>  and  \j/. 

(b)  If  <£  and  if/  are  any  two  conjugate  functions  of  x  and  y, 
and  if  a  and  (3  are  any  two  other  conjugate  functions  of  x 
and  y,  and  if  for  x  and  y  in  the  expressions  for  </>  and  ty  we 
substitute  the  expressions  for  a  and  (3,  we  shall  get  two  new 
conjugate  functions  of  x  and  y. 

(c)  If  <£i  and  fa,  </>2  and  \f/2  are  any  two  pairs  of  conjugate  func 
tions,  <^>!  ±  <£2  and  \pl  ±  ^2  are  conjugate  functions  of  x  and  y. 

257.  Prove  that  in  any  case  of  steady  uniplanar  flow  of 
electricity  —  that  is,  flow  which  at  every  point  is  parallel  to 
a  given  plane,  and  of  such  a  character  that  its  intensity  and 
direction  are  the  same  at  all  the  points  of  any  line  drawn  per 
pendicular  to  the  given  plane  —  there  exists  a  function  con 
jugate  to  the  potential  function.     This  function  is  called  the 
"  flow  function." 

258.  Show  by  the  ordinary  rules  for  treating  imaginary 
quantities  that,  if  z  =  x  -+-  yi,  z2,  V»,  log  z  will  yield  respec 
tively  the  following  pairs  of  conjugate  functions  :  A(x*  —  ?/2), 

n  a 

—  2Axy ;  Art  cos  -,  Art  sin  -  ;  A  log  r,  Ad ;  where  r2  =  x2  +  ?/2 

J  .4 

and  0  =  tan"1-.     State  some  problems  of  steady  flow  within 
x 

conductors  which  these  conjugate  functions  will  help  to  solve. 

259.  Show  that,  with  certain  broad  limitations,  either  one 
(say  <£)  of  any  pair  (<£,  ^)  of  conjugate  functions  of  x  and  y  may 
be  taken  as  the  potential  function  in  empty  space  due  to  an 
electrostatic  distribution  the  density  of  which  is  a  function  of  x 
and  y  only,  and  which,  therefore,  must  be  constant  at  all  points 
on  any  indefinitely  extended  line  drawn  perpendicular  to  the 


MISCELLANEOUS    PROBLEMS.  431 

plane  of  xy.  Show  also  that  in  the  case  of  the  same  distribution 
the  other  function  \f/  will  be  constant  along  any  line  of  force. 

260.  Show  that  either  one  (say  <£)  of  any  pair  (<£,  \j/)  of  con 
jugate  functions  of  x  and  y  may  be  taken  as  the  potential  func 
tion  inside  a  conductor  which  carries  a  steady  current  flowing 
at  every  point  in  a  direction  parallel  to  the  plane  of  xy,  and  the 
same  in  intensity  and  direction  at  all  points  of  any  line  drawn 
perpendicular  to  this  plane.     Show  that  in  this  case  the  other 
function  ^  will  be  constant  along  any  line  of  flow,  and  that  the 
two  equations  <£  =  c,  \f/  =  c'  represent  respectively,  if  c  and  c' 
are  parameters,  cylindrical  equipotential  surfaces  and  cylin 
drical  surfaces  of  flow.     If  ds  is  the  element  of  any  curve  AB 
in  the  plane  xy,  and  if  D^  is  the  derivative  of  <£  taken  in  the 
direction  of  the  normal  to  ds  which  points  towards  the  right  as 

CB 
one  goes  along  the  curve  from  A  to  B,  the  integral  —  k  I  Dn<f>  •  ds 

gives  the  amount  of  positive  electricity  which  crosses  per  unit 
of  time  from  left  to  right  so  much  of  a  right  cylindrical  surface 
erected  on  AB  as  is  enclosed  by  two  planes  parallel  to  the 
plane  of  xy  and  at  the  unit  distance  from  each  other.  Since 
Dn<f>  =  Ds  «/f,  the  integral  just  considered  is  equal  to  —  k({j/B  —  \f/A), 
and  —  k  times  the  difference  between  the  values  of  ^  on  two 
right  cylindrical  surfaces  of  flow  gives  the  amount  of  flow 
across  the  unit  height  of  so  much  of  any  cylindrical  surface 
which  cuts  the  plane  of  xy  at  right  angles  as  is  included 
between  the  given  surfaces  of  flow. 

261.  Prove  that: 

(a)  If  ?-!,  ?-2,  ?-3,  •  •  •,  rn  are  the  lengths  of  the  radii  vectores 
drawn  from  any  point  P  to  any  n  parallel  axes,  and  if  01? 
02,  Q3,  •  •  •,  6H  are  the  angles  which  these  radii  vectores  make  with 
a  fixed  line  in  the  plane  of  xy  which  is  perpendicular  to  the  axes, 

4>  =  Al  log?-!  +  A2  log 7-3  +  A3  log r3  + h  An  log rn, 

f  =  A&  +  A202  +  AS03  +  •  •  •  +  AnOn 
are  conjugate  solutions  of  Laplace's  Equation. 


432  MISCELLANEOUS    PROBLEMS. 

(b)  The  equation  \j/  =  c'  represents  for  each  value  of  c'  a 
cylindrical  surface  which  passes  through  all  the  axes. 

(c)  For  very  large  values  of  c,  the  equation  <£  =  c  represents 
as  many  closed  cylindrical  surfaces,  each  surrounding  one  of 
the  axes,  as  there  are  positive  terms  in  the  expression  for  <£. 

(d)  For  very  large  negative  values  of  c,  the  equation  <£  =  c 
represents  as  many  closed  cylindrical  surfaces,  each  surround 
ing  one  of  the  axes,  as  there  are  negative  terms  in  <£. 

(e)  If  %A  =  0,  no  one  of  the  cylindrical  surfaces  \f/  =  c'  ends 
at  infinity. 

(/)  The  value  of  (  Dn<f>  •  ds  taken  around  any  closed  curve 

in  the  plane  xy  which  surrounds  the  jth  axis  and  no  other  is 
equal  to  the  change  made  in  \f/  by  going  around  the  curve,  and 
this  is  2irAj. 

(g)  However  the  axes  may  be  distributed  and  whatever 
values  may  be  assigned  to  the  A's,  <f>  represents  the  potential 
function  corresponding  to  a  uniplanar  flow  of  electricity* 
within  the  substance  of  an  infinite  conducting  lamina,  either 
thick  or  thin,  when  cylindrical  holes,  on  the  curved  surface  of 
each  one  of  which  <f>  is  constant,  are  cut  through  the  lamina 
so  as  to  remove  all  the  axes,  and  if  the  curved  surfaces  of 
these  holes  are  kept  at  potentials  equal  to  the  values  of  <£  on 
them.  This  is  practically  the  case  of  a  very  large  thin  sheet 
of  metal  touched  at  certain  points  by  the  ends  of  wires  con 
nected  with  the  poles  of  batteries. 

(A)  If  in  the  value  of  <£  there  is  an  even  number  (2m)  of 
terms,  half  of  which  are  positive  and  half  negative,  and  if, 
moreover,  all  the  A's  are  numerically  equal,  we  have  the  case 
in  which  m  similar  pieces  of  wire  connected  with  the  positive 
pole  of  a  battery  touch  a  thin  sheet  of  metal  in  in  places,  and 
m  similar  pieces  of  wire  connected  with  the  negative  pole  of 


*  See  papers  by  Foster  and  Lodge  in  the  Philosophical  Magazine  for 
1875  and  1876. 


MISCELLANEOUS    PROBLEMS.  433 

the  battery  touch  the  metallic  sheet  in  m  other  places.  In 
this  case,  if  PI  and  P2  are  any  two  points  in  the  nietal,  the 
resistance  of  so  much  of  the  sheet  as  lies  between  the  equipo- 

T*/*      -    T*P 

tential  surfaces  on  which  Pl  and  P2  lie  is   ..  2   .    7  *»  when  8  is 


the  thickness  of  the  lamina,  and  k  its  specific  conductivity. 

(i)  If  <f>  consists  of  two  terms  the  coefficients  of  which  are 
numerically  equal  but  opposite  in  sign,  we  have  the  case  of  a 
thin  sheet  of  metal  touched  at  two  points  by  the  two  poles  of 
a  battery.  Here  the  curves  in  the  plane  xy,  for  which  i//  is 
constant,  are  circles  (Fig.  59)  the  centres  of  which  are  on  the 
line  which  bisects  at  right  angles  the  line  which  gives  the 
points  where  the  battery  electrodes  touch  the  sheet. 

Show  that  this  value  of  <£  enables  us  to  find  the  resistance 
of  a  thin  circular  disc  touched  at  two  points  on  its  circumfer 
ence  by  the  poles  of  a  battery,  and  hence,  by  superposition, 
the  resistance  of  such  a  disc  touched  by  any  number  of  pairs 
of  battery  poles  at  different  places  on  the  circumference. 
State  other  problems  which  an  inspection  of  Fig.  59  shows 
can  be  solved  by  the  aid  of  the  value  of  c£. 

(j)  If  <f>  is  made  up  of  an  infinite  number  of  terms  with 
coefficients  all  numerically  equal,  but  alternately  positive  and 
negative,  and  if  the  corresponding  axes  cut  the  plane  of  yy  in 
a  straight  line  so  that  the  distance  between  any  axis  and  the 
next  is  £,  certain  of  the  lines  of  force  in  the  plane  of  xy  will 
be  straight  lines  which  cut  at  right  angles  the  line  on  which 
the  traces  of  the  axes  lie.  Show  that  by  aid  of  this  <f>  we  can 
find  the  resistance  of  a  lamina  of  breadth  b,  and  of  infinite 
length  when  touched  at  two  points  opposite  each  other,  one 
on  one  edge,  and  the  other  on  the  other.  Draw  from  general 
knowledge  a  diagram  which  shall  give  the  shape  of  the  lines 
of  flow  and  the  equipotential  lines  in  such  a  lamina. 

262.  (a)  Show  that  if  in  a  thin  conducting  plate  of  indefi 
nite  extent  there  are  two  sources  and  a  sink,  each  of  strength 


434  MISCELLANEOUS    PROBLEMS. 

numerically  equal  to  m,  situated  respectively  at  points  A,  B, 
C,  which  lie  in  order  upon  a  straight  line,  one  of  the  lines  of 
flow  consists  in  part  of  a  circumference  of  radius  V  CA  •  CB 
drawn  around  C  as  a  centre,  so  that  the  flow  inside  the 
circumference  would  be  unchanged  if  the  part  of  the  plate 
outside  it  were  cut  away.  In  other  words,  if  a  circumference 
be  drawn  in  a  thin  conducting  plane  plate  of  indefinite  extent, 
the  "  image  "  in  this  circumference  of  a  source,  of  strength  m, 
situated  at  a  point  P  in  the  plane,  is  made  up  of  a  sink,  of 
strength  w,  at  the  centre  of  the  circle,  and  a  source  of  the 
same  strength  at  Q,  the  inverse  point  of  P  with  respect  to 
the  circumference. 

Show  that  if  a  sink  be  regarded  as  a  negative  source,  and  if, 
inside  a  circumference  drawn  in  a  thin  plane  conducting  plate 
of  indefinite  extent,  there  are  sources  at  the  points  A1}  A2, 
A3,  •••,  Ak,  of  strengths  algebraically  equal  to  mlf  ra2,  ms, 
-•,mk  respectively,  and  sources  of  strengths  algebraically 
equal  to  —  mlt  —  m2,  —  ms,  •••,  —  mk,  at  the  corresponding 
inverse  points,  then,  if  mx  4-  ra2  +  ms  +  •  •  •  +  mk  =  0,  there 
is  no  flow  of  electricity  across  the  circumference. 

If  at  a  fixed  point  P  in  a  thin  plane  plate  (Fig.  127)  there 

is  a  sink  of  strength  numerically  equal  to  m,  and  at  another 

p  A     point  P'  in  the  plate  an  equal  source,  and  if  P' 

^^  be  made  to  approach  P  as  a  limit  and  the  prod- 

*  *    /        uct  m  •  PP'  be  kept  always  equal  to  a  given  con- 
Ofa — X   stant  fjL,  we  have  as  a  limit  a  "plane  doublet"* 

of  strength  /x,  the  axis  of  which  is  PX,  the  limit 
ing  position  of  the  straight  line  drawn  from  P 
to  P'.     We  shall  find  it  convenient  to  represent  sources  and 
sinks  respectively  by  black  and  unshaded  circles,  and  doublets 
by  circles  half  black  and  half  unshaded.     The  black  portion 

*  Kirchhoff,  Pogg.  Ann.,  1845,  p.  497.     W.  R.  Smith,  Proc.  Ed.  Eoy. 
Soc.,  1869-70.    Foster  and  Lodge,  Phil.  Mag.,  1875.  Minchin's  Uniplanar 
Kinematics,  p.  213.     Peirce,  Proc.  Am.  Acad.  of  Arts  and  Sciences,  1891. 


MISCELLANEOUS    PROBLEMS. 


435 


of  a  doublet  circle  indicates  the  directions  in  which  there  is 
a  flow  away  from  the  point  where  the  doublet  is  situated; 
the  unshaded  portion  indicates  the  directions  from  which 
there  is  a  flow  towards  this  point.  The  axis  of  a  doublet 
bisects  both  the  black  and  unshaded  portions  of  the  doublet 
circle.  Show  that  if  P  be  used  as  origin  and  PX  as  axis 
of  abscissas,  the  velocity  potential  function  due  to  the  doublet 


s   <    =  — 


and  the 


function  is      = 


If 


*2  +  r'  *2  +  r 

x  +  yi  =  z,  these  are  respectively  the  real  part  and  the  real 
factor  of   the    imaginary   part   of   the   function    :— •     The 

equipotential    lines  and   the  lines   of   flow    are   circles   (see 
Fig.  128)  touching  the  axes  of  y 
and  x  respectively  at  the  origin. 

A  "plane  quadruplet"  is 
formed  of  two  equal  and  oppo 
site  plane  doublets  in  the  same 
manner  that  a  doublet  is  formed 
out  of  a  source  and  an  equal 
sink.  An  "  octuplet "  is  formed 
in  a  similar  way  of  two  equal 
and  opposite  quadruplets,  and 
so  on.  We  may  use  the  word 
"  motor  "  to  denote  in  general  a 
source,  a  sink,  a  doublet,  a  quad 
ruplet,  or  any  other  combination  of  sources  or  sinks  at  a 
single  point. 

(b)  The  upper  circle  in  Fig.  130  shows  the  plane  quadru 
plet  formed  by  combining  the  two  plane  doublets  indicated  in 
the  lower  part  of  this  diagram.  Show  that  the  flow  function  due 
to  a  quadruplet  of  this  kind  at  the  origin  is  —  2  kxy  /  (x*  +  ?/2)2, 
while  the  flow  function  due  to  such  a  quadruplet  as  that 
shown  in  Fig.  131  will  be  k(x2  -  if}/(y?  +  y2)2.  One  of 
these  quadruplets  is  evidently  equivalent  to  the  other  turned 


FIG.  128. 


436  MISCELLANEOUS    PROBLEMS. 

through  45°.  Find  the  flow  function  clue  to  an  octuplet  of 
the  kind  shown  in  Fig.  132  at  the  origin. 

(c)  Show  that  the  lines  of  flow  due  to  a  plane  doublet  may 
be  regarded  as  the  lines  of  force  due  to  a  columnar  magnet  of 
infinitely  small  cross-section. 

(d)  Show  that  the  functions 

112         6 
te^->-yy--y- 

each  of  which  is  the  derivative  with  respect  to  z  of  the  one 
which  precedes  it,  yield  a  series  of  pairs  of  conjugate  func 
tions  which  represent  in  order  the  velocity  potential  functions 
and  the  flow  functions  due  to  a  source  at  the  origin,  to  a  plane 


€          e 

€>  < 


FIG.  129.  FIG.  130.  FIG.  131.  FIG.  132. 

doublet  at  the  origin,  to  a  plane  quadruplet  at  the  origin,  to  a 
plane  octuplet  at  the  origin,  and  so  on. 

(e)  Show  that  if  two  plane  doublets  L  and  M  exist  together 
at  a  point  0,  and  if  the  directions  of  the  two  straight  lines  OA, 
OB  show  the  directions  of  the  axes  of  L  and  M  respectively, 
and  the  lengths  of  OA  and  OB  the  strengths  of  L  and  M  on 
some  convenient  scale,  then  the  direction  of  the  axis  of  the 
resultant  of  L  and  M  will  be  given  by  the  direction,  and  the 
strength  of  the  resultant  by  the  length,  of  the  diagonal  of 
the  parallelogram  of  which  OA  and  OB  are  adjacent  sides. 
Plane  doublets,  then,  can  be  compounded  and  resolved  by  com 
pounding  and  resolving  their  axes  like  forces  or  velocities. 

263.  If  a  charge  -f  m  concentrated  at  a  point  Q  be  made  to 
approach  on  any  analytic  curve  a  point  charge  —  m  at  a  fixed 
point  P  011  the  curve,  and  if  as  Q  approaches  P,  m  is  made  to 
increase  in  such  a  manner  that  the  product  of  m  and  PQ  is 


MISCELLANEOUS    PROBLEMS.  437 

always  equal  to  the  constant  p,  the  limiting  value  of  the  poten 
tial  function  of  the  system  is  said  to  be  due  to  a  space  doublet 
of  strength  /A  at  the  point  P,  and  the  axis  of  the  doublet  is 
said  to  be  the  limiting  position  of  the  secant  PQ.  Show 
that  if  r  is  the  distance  of  any  point  P'  from  P  and  if  0  is 
the  angle  between  the  axis  of  the  doublet  and  PP't  the  value 
at  P'  of  the  potential  function  due  to  the  doublet  is  /*cos  Q  /i*. 

The  force  components  along  and  perpendicular  to  r  are 
2/A  cos  O/ 1*  and  /t  sin  0 / r3.  The  potential  function  (Section  69) 
due  to  a  doublet  at  the  origin  with  axis  coincident  with  the 
x  axis  is  px/  r3. 

The  potential  function  due  to  a  mass  —  m  at  the  point  (b,  0, 0), 
a  mass  -f  m  at  the  point  (b  +  8,  0,  0),  a  mass  —  ma/  (b  +  8)  at 
the  point  (a? / (b  -f  8),  0,  0),  and  a  mass  ma/b  at  the  point 
(a2/ b,  0,  0),  where  b  and  8  are  smaller  than  «,  has  the  value 
zero  on  the  spherical  surface  x-  +  if  -f-  z-  =  a2.  Prove  that  if, 
while  a  and  b  are  constant,  8  be  made  to  decrease  indefinitely 
and  m  to  increase  in  such  a  manner  that  their  product  shall 
always  be  equal  to  the  given  constant  p.,  the  limiting  value 
of  the  potential  function  will  be 


(x  -  b)/[(x  -  b)°  +  if  +          +  ap  [l(x~  +  if  +  22) 

-  «2)2  +  62  (/  -f 


If  6  =  0,  this  expression  becomes  px  (a3  "~  y'3)  /a3'"3?  where 
r2  —  x2  -f-  ^  H-  22.  What  problem  in  electrostatics  can  be  solved 
by  the  aid  of  this  last  function  ?  Is  the  image  of  a  doublet 
in  a  spherical  surface  another  doublet  ? 

264.  A  straight  wire  of  radius  a  which  forms  the  core  of  a 
cable  of  length  I  lies  in  the  axis  of  x  with  one  end  at  the 
origin  and  the  other  at  the  point  (7,  0,  0).  The  whole  of  the 
outside  of  the  insulating  covering  of  the  cable  and  the  core 
at  the  point  (I,  0,  0)  are  kept  at  potential  zero,  while  the  core 
at  the  origin  is  at  the  potential  F0.  Show  that  if  c  is 
the  capacity  per  unit  length  of  the  cable  considered  as  a 


438  MISCELLANEOUS    PROBLEMS. 

condenser,  k  the  ratio  of  the  conductivity  per  unit  length 
of  the  core  to  c,  and  h  the  rate  of  loss  of  electricity  by  leak 
age  through  the  insulation  per  unit  length  of  the  cable  when 
the  difference  of  potential  of  the  core  and  the  outside  of  the 
cable  is  unity,  the  value  of  the  potential  function  V  in  the 
core  satisfies  the  equation 

Dt  F=  k  •  D*  V-  ~  V,  and  if  V  =  we-ht/c,  D{w  =  k-  Dx2w  . 
c 

Show  also  that  in  the  final  state,  when  V  satisfies  the  equa 
tion  D*V  =  hV/kc  and  is  equal  to  F0  when  x  =  0,  and  to 
zero  when  x  =  I,  the  value  of  V  is  given  by  the  expression 
[  Fr  sinh  /3a;  +  Fe-smh/3(Z  —  £e)]/sinh/3Z,  where  (32  =  h/kc. 
Prove  that  any  quantity  of  the  form  As-  e~xt  cos(nx  —  8), 
where  X  =  kn2,  satisfies  the  equation  Dtw  =  k  •  Dx2w,  and  that 
if  8  =  J-TT,  n  =  sir  /I,  where  s  is  an  integer,  and 

As  =  —  2  ckirs  -  COS  Sir  •  /  (hi2  +  cksW)  ; 
the  expression 

wl  =  N    A8e~Kt  sin  TIX  vanishes  when  jc  =  0  or  sc  =  Z, 

s=l 

and,  when  £  =  0,  is  equal  to  —  sinh  ft  (I  —  x)  /  sinli  pi.  Hence 
prove  that  the  expression 

V=  F0[sinh  ft  (I  -  a)  /sinh  ftl  -  w^~u'^ 

gives  the  value  of  the  potential  function  in  the  cable,  if, 
when  the  whole  core  is  at  potential  zero  and  the  farther  end 
permanently  grounded,  the  point  x  =  0  is  suddenly  raised  to 
potential  F0  at  the  time  t  =  0,  and  kept  there.  The  current 
(C)  at  any  point  is  given  by  the  negative  of  the  derivative  of 
the  potential  function  with  respect  to  x,  divided  by  the  resist 
ance  p  of  the  core  per  unit  length,  so  that 


"  cos  nx]. 


MISCELLANEOUS    PROBLEMS.  439 

If  the  insulation  is  so  good  that  h  may  be  neglected, 


and  the  current  is 

(Fo/p/X1  +  2-COSS7T.  V"e~A'  COSTIX). 

265.  The  terminals  of  a  battery  of  electromotive  force  EQ 
volts  and  internal  resistance  b  ohms  are  suddenly  connected, 
through  a  non-inductive  conductor  of  resistance  r  —  b  ohms, 
with  the  coatings  of  a  condenser  of  k  farads  capacity.     Show 
that  after  t  seconds  the  condenser  is  charged  to  potential 
difference  E  volts,  where  E  =  E(,(l  -  e~(  'kr)  =  E0T,  and  that 
the  charge  on  the  positive  plate  is  Ek  units.     If  t  =  ^  kr, 
T  =  0.095  ;  if  t  =  J-  kr,  T  =  0.181  ;   if  t  =  J  kr,  T  =  0.393  ;   if 
t  =  kr,  T=  0.632  ;  if  t  =  2kr,  T-  0.865  ;  if  t  =  3kr,  T=  0.950  ; 
if  t  =  5  kr,  T  =  0.993,  and  if  t  =  7  kr,  T  =  0.999. 

Show  that  if  the  condenser  just  mentioned  had  been  leaky, 
its  dielectric  having  a  resistance  of  only  R  ohms,  the  charge 
on  the  positive  coating  after  t  seconds  would  have  been 

^TT5(1  -  e-'<r  +  *>/*»*),  and  the  final  charge  EQkR  /  (r  +  K). 

266.  The  coatings  of  a  perfect  condenser  of  2  microfarads 
capacity  which  are  connected   together   by  a  non-inductive 
resistance  R  of  2500  ohms  are  attached  to  the  terminals  of 
a  constant  battery.     After  the  condenser  has  become   fully 
charged,  a  bullet  moving  at  a  velocity  of  v  metres  per  second 
cuts  first  one  of  the  battery  leads  at  a  point  A  and,  2  metres 
farther  on  in  its  course,  the  resistance  R  at  a  point  B.   While 
the  bullet  is  moving  from  A  to  B  the  condenser  loses  1  —  1/e 
of  its  charge  through  R.     Show  that,  e  being  the  base  of  the 
natural  system  of  logarithms,  v  =  400. 

267.  If  Si  and  Sz,  the  plates  of  a  condenser  separated  by  a 
poorly  conducting  medium  of  inductivity  /x  and  of  conductivity 
X,  are  at  potentials  Vl  and  V%  respectively,  and  if  V  denotes 


440  MISCELLANEOUS    PROBLEMS. 

the  potential  function  in  the  dielectric,  the  capacity  of  the  con 
denser  and  the  strength  of  the  current  that  flows  through  the 
dielectric,  when  the  difference  of  potential  of  the  plates  is 
unity,  are 


Show  that  if  the  condenser  be  charged  to  such  a  potential 
that  each  plate  requires  Q0  units  of  (positive  or  negative) 
electricity  and  then,  left  to  itself,  the  charge  on  one  of  the 
plates  after  t  seconds  is  given  numerically  by  the  expression 
Q**Q0-+*». 

268.  A  Leyden  jar  loses  0.000001  of  its  charge  per  second  by 
conduction  through  the  glass.     The  specific  inductive  capacity 
of  the  glass  is  8.     Show  that  the  resistance  of  a  cubic  centi 
metre  of  the  glass  is  roughly  14  X  1017  ohms,  having  given 
that   one   electrostatic   unit  of   resistance    is    equivalent   to 
9  X  1011  ohms.     [M.  T.] 

269.  A  submarine  telegraph  cable  1885  miles  long  is  formed 

of  a  copper  conductor  —  inches  in  diameter  surrounded  by  a 

gutta-percha  coating  £  inch  in  diameter.  The  specific  inductive 
capacity  of  gutta-percha  being  4.2,  show  that  the  capacity  of 
the  cable  is  equal  to  that  of  a  sphere  of  the  same  size  as  the 
earth.  [St.  John's  College.] 

270.  The  outer  coatings  of  two  condensers  A  and  B  are  put 
to  earth  and  their  inner  coatings  are  connected  through  a 
galvanometer  the  resistance  of  which  is  4000  ohms.     The 
capacity  of  A  is  3  microfarads,  that  of  B  is  1  microfarad,  and 
the  two  condensers  are  charged  to  potential   1  volt.     The 
inner  coatings  of  A  and  B  are  then  put  to  earth  simultane 
ously  through  resistances  of  1000  and  2000  ohms  respectively. 
Show  that  the  whole  amount  of  electricity  which  will  flow 
through  the  galvanometer  is  one-seventh  of  the  charge  of  the 
smaller  condenser.     [St.  John's  College.] 


MISCELLANEOUS    PROBLEMS.  441 

271.  The  outer  coatings  of  two  condensers  A  and  B  are 
put  to  earth  and  their  inner  coatings  are  connected  together 
through  a  galvanometer  of  g  ohms  resistance.  The  capaci 
ties  of  the  condensers  are  C  and  c  respectively.  Both  are 
charged  initially  to  potential  F"0  and  then  have  charges  QQ 
and  q0.  Show  that  if  the  inner  coatings  of  the  condensers  are 
put  to  earth  simultaneously  through  non-inductive  resistances 
R  and  r,  and  if 

X=rRC,    X'=rRC)    p.  =  Cr(g  +  E),    p'  =  CR(g  +  r\ 
m  =  CcrRg,  k2  =  4  AA'+  (/i-/)2;  ^  -  XX'  =  CcrRff  (y  +  r  +  fi), 
and  the  charge  on  A  after  t  seconds  will  be 

Q0e-^+  »0'/  2«[(A;  +  p  +  ,j,'  -  2  m/  CR)  <F  /2'" 

+  (k-n-iL'  +  2m/  CR)  e-kt/  2m]  /2  k. 

Show  also  that  the  whole  quantity  of  electricity  which  passes 
through  the  galvanometer  during  the  discharge  is 


272.  Prove  that  the  potential  and  stream  line  functions  due 
to  electrodes  placed  at  certain  points  of  a  spherical  current 
sheet  can  be  deduced  directly  from  the  solutions  for  the  plane 
current  sheet  which  is  its  stereographic  projection.     If  El  and 
E2  be  two  electrodes  on  a  complete  spherical  sheet,  show  that 
the  stream  lines  are  small  circles  through  El  and  E2  and  the 
equipotential  curves  small  circles  the  planes  of  which  pass 
through  the  line  of  intersection  of  the  tangent  planes  at  E± 
and  E2. 

273.  Verify  the  statement  that  the  value  of  the  potential 
function  at  any  point  P  of  a  solid  homogeneous  sphere  of 
specific  resistance   K,  when  a  current  of  intensity    C  flows 
between   two   electrodes   A   and   B   at   opposite    ends    of   a 
diameter,  is 

—      — 

BP      AB    ° 


442  MISCELLANEOUS   PROBLEMS. 

where  N  is  the  foot  of  the  perpendicular  from  P  on  the 
diameter  AB.     [M.  T.] 

274.  The  two  concentric  spherical  surfaces  which  bound  a 
shell  are  kept  at  different  constant  potentials.     Prove  that  if 
the  conductivity  of  the  shell  is  a  function  of  the  distance 
from  its  centre,  the  potential  function  within  it  satisfies  the 
equation*  Dr(r*k-DrV)  =  0.     Show  that  if  w  =  l/r,  this  is 
equivalent  to  the  equation  given  on  page  250. 

275.  Prove  that  if  a  quantity  of  electricity  equivalent  to  Q 
absolute  electromagnetic  units  be  discharged  through  a  ballistic 
galvanometer  which  has  a  suspended  system  the  magnetic 
moment  of  which  is  M,  the  moment  of  inertia  /,  and  the 
reduced  complete  time  of  swing  T0, 

47T/ 


where  GM  is  the  couple  exerted  upon  the  suspended  system 
in  its  position  of  equilibrium  when  a  steady  current  of  1 
unit  passes  through  the  galvanometer  coil. 

276.  When  a  bar  magnet  of  magnetic  length  2  1  and  moment 
M  is  placed  in  Gauss's  A  position  with  its  centre  at  a  dis 
tance  d  from  the  centre  of  a  magnetic  needle  of  length  2  A, 
the  needle  is  deflected  through  an  angle  a,  such  that 

2Hi2M.ad-l      d—l      d  +  l      d  +  l 

' 


M 

where  r-f  =  (d  —  I  —  A  sin  a)2  +  A2  cos2  a, 

r22  =  (d  —  I  +  A  sin  a)2  +  A2  cos2  a, 
r,a  =  (d  +  I  -  A  sin  a)2  +  A2  cos2  a, 
r£  =  (d  +  I  +  A  sin  a)2  +  A2  cos2  a. 

Show  that  if  1  =  4:  centimetres,  d  =  40  centimetres,  A  =  0.5 
centimetre,  a  =  20°,  and  H=  0.2,  this  formula  makes  m  =  285.43, 
whereas  the  approximate  formula, 

M      t,n  —  /2\a 

m_  =  i^.^J_  tan  a)  yields  m  =  285.40. 

//  +j  d 


MISCELLANEOUS    PROBLEMS.  443 

277.  A  magnetometer  is  set  up  with  the  centre  of  its  needle 
vertically  above  a  point  in  the  axis  of  a  horizontal  metre  rod 
n  centimeters  from  the  centre.     The  rod  is  perpendicular  to 
the  meridian.     A  homogeneous,  short  bar  magnet  is  placed  in 
Gauss's  A  position  with  its  centre  first  50  —  d  centimetres 
from  one  end  of  the  rod  and  then  50  —  d  centimetres  from  the 
other  end,  d  being  greater  than  n.     If  the  deflections  of  the 
magnetometer  needle  in  the  two  cases  are  8j  and  82  respec 
tively,  the  relative  error  made  by  computing  M/H  by  means 
of  the  formula 

d3(tan  3L  +  tan  82)  /4  is  [(1  +  3  ^2)  /  (1  -  e2)3]  -  1, 

where  e  —  n/d. 

278.  The  track  upon  which  the  carriage  of  the  short  deflect 
ing  magnet  slides  in  an  apparatus  for  determining  M/H  in 
Gauss's  A  position  makes  an  angle  0  with  the  east  and  west 
line  instead  of  being  exactly  perpendicular  to  the  meridian. 
Show  that  if  the  centre  of  the  deflecting  magnet  is  at  a  dis 
tance  d  from  the  centre  of  the  needle,  and  if  the  deflection 
changes  from  Sl  to  —  82  when  the  deflector  is  turned  end 
for  end, 


H      2  cos  (Si  +  (9)      2  cos  (82  -  0) 

.      ctn  8l  —  ctn  82 
where  tan  6  = —^ 

2i 

279.  In  order  to  obtain  the  temperature  coefficient  of  a  cer 
tain  magnet,  of  moment  Mlt  it  is  placed  in  a  water  bath  at 
a  short  distance  from  a  magnetometer  needle,  its  axis  being 
perpendicular  to  the  magnetic  meridian  at  the  centre  of  the 
needle.  The  needle  is  brought  back  to  its  zero  position  by  a 
compensating  magnet  placed  on  the  opposite  side  of  the 
magnetometer  at  a  distance  d0  from  it,  its  axis  being  also 
perpendicular  to  the  meridian  at  the  centre  of  the  needle. 
The  moment  of  the  compensating  magnet  is  Mot  its  magnetic 


444  MISCELLANEOUS    PROBLEMS. 

length  2  10  .  When  the  magnet  Ml  is  heated  a  given  number 
of  degrees,  its  moment  decreases  to  Jf/,  and  the  magnet 
ometer  needle  is  deflected  over  n  divisions  of  the  scale.  The 
scale  distance  being  a,  prove  that 


where  the  deflection  n  is  small. 

Show  that  if  ax  is  the  angle  through  which  M0  would  deflect 
the  needle  if  Ml  were  absent, 

M-,  —  M,  f       tan  a  n 

—  —  —  —  -  =  -  j  where  tan  a  =  7—  • 
Ml  tan  c^  2  a 

280.  Two  magnets,  m^  and  ra2,  are  placed,  with  their 
axes  parallel  to  each  other  but  opposite  in  direction,  in 
Gauss's  B  position  with  respect  to  a  magnetometer.  The 
centre  of  ml  is  north  of  the  magnetometer  and  the  centre  of 
mz  south  of  it.  The  distances  (dl  and  d2)  of  the  centres  of 
w&i  and  ra2  from  the  centre  of  the  magnetometer  needle  are 
such  that  the  needle  is  undeflected.  Show  that  if  fa  and  p2 
are  the  strengths  of  the  "poles"  of  ml  and  m2J  and  if  211}  212, 
and  2  A.  are  the  "lengths"  of  m^  m2,  and  the  needle  respec 
tively,  /AJ  is  to  fji2  as 

J  _  r       , 

'2  I  [42  +  W  -  *)2]»        [^  +  ( 


' 


281.  A  fixed  bar  magnet  of  magnetic  length  SN=  2  L  and 
of  pole  strength  M,  and  a  magnetic  needle  of  magnetic  length 
sn  =  21,  of  pole  strength  w,  are  in  the  same  plane,  with  their 
centres  (C,  c}  at  a  distance  r  from  each  other.  The  angles 
NCc  and  scC  are  equal  to  $  and  <£  respectively.  The  lines 


MISCELLANEOUS    PROBLEMS.  445 

ns,  £Ar  meet  when  produced  in  F.  The  perpendicular  dis 
tances  of  Nj  C,  and  S  from  ns  are 

r  sin  <£  —  L  sin  (<£  -f  <£),  r  sin  <£,  and  ?•  sin  <f>  +  L  sin  (<£  +  <J>), 

so  that  the  length  of  the  perpendicular  dropped  from  c  upon 
Nn  is  I  sin  (FnN)  or  Z  [>•  sin  <£  —  Z  sin  (<£  +  <£)]  /  Nn.  The 
lengths  of  the  perpendiculars  dropped  from  c  upon  Sn,  Ns, 
and  /Ss  are 

I  [>'  sin  <£  +  L  sin  (<£  +  <£)]  /  Sn,  I  \r  sin  <£  —  L  sin  (<£  +  <£)]  /  JVs, 
and  I  [>•  sin  <£  +  £  sin  (<£ 


Show  that  the  sum  of  the  moments,  taken  about  c,  of  the 
forces  which  tend  to  decrease  <£,  is 


D  =  Mm     J  =rr^  +  =z  >  ^  r  sin  <f>  —  L  sin  (<f>  4- 

Ll-ar*     jvvj    L 


in 

^ 
jj 


or 


J  r'rin  4-  1  =i  +  =  r-   =  ~  = 

[  |_3?i        ^j?        Sn        bs  J 


Show  also  that  Nn2  =  r  +  L2  +  f2  +  2  >•/  cos  ^>  —  2  ?-Z  cos 

-  2  Zi  cos    ^  4-  *,  or  =  -         +  2/cos  ^  ~  2L  cos 

r 


2"|~g 

, 


and  that,  if  both  I  and  L 

are  small  compared  with   r  so  that  only  the  first   powers 
of     l/r    and   L/r  need    be    kept,    the    approximate    value 

1  f-i  j_  3  (^  cos  ^  -  7  cos  $)~|  -— 

^    1  -h  -  —  may  be  used  for  ^  n~3.     Treat 

ing  JVs,  Sn,  and  /Ss  in  the  same  way,  prove  that  if  J/0  and 


440  MISCELLANEOUS    PROBLEMS. 

m0  are  the  magnetic  moments  of  the  magnet  and  the  needle 
respectively,  we  may  use  for  D  the  approximate  value 

MQ  m0  [3  cos  <£  sin  </>  —  sin  (<£  +  <fr)]  /r\ 

A  better  approximation  can  be  obtained  by  keeping  higher 
powers  of  the  ratios  //rand  L/r.  It  is  to  be  noticed  that 
<£  =  0  and  <fc  =  90°  correspond  to  Gauss's  "  Principal  Positions." 

282.  Prove  that  the  magnetic  force  at  a  large  distance  in 
the  prolongation  of  its  axis,  due  to  a  bar  magnet  of  moment 
My  lies  between  2  M/rf  and  2  M/rf,   where  rlt  rz  are  the 
distances  from  the  two  ends  of  the  magnet. 

283.  If  I,  m,  n  are  the  direction  cosines  of  the  axis  of  a 
small  magnetic  needle  free  to  turn  about  its  centre  in  a  mag 
netic  field,  and  if  //,  M,  N  are  the  components  of  the  couple 
which  acts  on  the  needle,  DtL  -f  DmM+  DnN=  0. 

284.  The  accurately  flat  north  end  of  one  of  two  exactly 
similar,  uniformly  polarized,  perfectly  hard  bar  magnets  is 
placed  in  close  contact  with  the  south  end  of  the  other,  so 
that  the  two  form  a  long,  uniformly  polarized,  straight  bar. 
What  force  is  necessary  to  separate  the  magnets  lengthwise  ? 
Compute  the  work  necessary  to  separate  into  short  elements 
a  long,  uniformly  polarized,  magnetic  filament. 

285.  Has  a  polarized  rigid  distribution  an  axis  in  the  sense 
that  a  straight  bar  magnet  has  a  magnetic  axis  ?     Consider 
first  a  bent,  solenoidally  polarized,  magnetic  filament. 

286.  Show  that  if  a  polarized  electrical  distribution  were 
enclosed  in  a  thin  "  metallic  skin  connected  with  the  earth," 
there  would  be  induced  upon  the  inner  surface  of  the  skin  a 
charge,  E,  of  total  amount  zero.     Show  also  that  the  effect  of 
the  given  distribution  together  with  the  charge  on  the  inner 
surface  of  the  skin  would  be  nothing  at  outside  points,  and 
the  effect  at  outside  points  of  the  given  distribution  the  same 
as  that  of  a  charge  on  the  skin  equal  to  the  negative  of  E. 
This  charge  is  sometimes  called  "Green's  Distribution"  and 
sometimes  "Poisson's  Surface  Distribution." 


MISCELLANEOUS    PROBLEMS.  447 

287.  A  solid  soft-iron  sphere  is  placed  in  a  uniform  magnetic 
field.     Show  that  about  three  times  as  many  lines  of  force 
pass  through  any  closed  curve  within  the  sphere  as  through 
an  equal  and  parallel  curve  at  an  infinite  distance. 

288.  In  the  case  of  a  certain  sphere  of  radius  a  polarized 
parallel  to  the  axis  of  x,  I  =  A0-  r"-3,  where  r  is  the  distance 
from  the  centre,  and  the  function,  /,  mentioned  on  page  192, 
is  r"-*.    Show  that  the  values  of  the  potential  function  within 
and  without  the  sphere  are 

4ir^0xr"~3/;i  and  4  TrA^x/nr*  respectively. 

Show  that  at  the  surface  of  the  sphere  the  normal  component 
of  the  induction  is  continuous,  and  the  tangential  components 
in  general  discontinuous.  The  tangential  components  of  the 
force  are  continuous,  and  the  normal  component  in  general 
discontinuous,  by  the  amount  47ro-. 

289.  An  uncharged  conducting  sphere  of  radius  a  is  in  a 
uniform  field  of  force  F,  and  consists  of  two  hemispheres  in 
contact  with  the  plane  of  division  perpendicular  to  the  field. 
Show  that  if  the  hemispheres  are  separated,  each  will  have  a 
charge  3a*F/±ir. 

290.  The  field  inside  a  shell  bounded  by.  two  concentric 
spherical  surfaces  of  radii  a  and  b  and  uniformly  polarized  in 
the  direction  of  the  x  axis,  has  the  potential  function  zero.  Out 
side  the  shell  the  potential  function  is  4  irxl(l*  —  a^/Sr8. 

291.  Show  that  -  3Xx/(p  +  2)  +  <7,  for  values  of  r  less 
than  a,  and  -  Xx  +  a?Xx(p  -  l)/[r3(/x  +  2)]  +  C,  for  values 
of  r  greater  than  o,  represent  the  potential  function  within 
and  without  a  sphere,  of  radius  a,  with  centre  at  the  origin, 
composed  of  a  homogeneous  dielectric  of  inductivity  /tx,  placed 
in  a  uniform  field  in  air  of  intensity  X. 

292.  If  a  cylindrical  surface  which  circumscribes  an  oval 
body  P  touches  it  in  a  curve  which  is  the  perimeter  of  a 
right  section  of  the  cylinder  of  area  Q,  and  if  P  be  uniformly 
polarized   in   the  direction  of  the  axis  of  the   cylinder   to 


448  MISCELLANEOUS    PROBLEMS. 

intensity  /,  the  amount  of  matter  in  the  positive  distribution 
on  P's  surface  is  Q  •  T.  If  an  ellipsoid  the  semiaxes  of  which 
are  a,  b,  c,  be  uniformly  polarized  to  intensity  /  in  the  direction 
of  the  axis  a,  the  moment,  M,  of  the  distribution  is  ^irdbcl  and 
the  amount  of  matter,  m,  on  either  the  positive  or  negative 
half  is  irbc  I.  The  distance  of  the  centre  of  gravity  of  either 
the  positive  or  negative  part  of  the  distribution  from  the 
centre  of  the  ellipsoid  is  M/2  m  or  f  a.  In  what  sense  is 
the  "  magnetic  length"  of  a  uniformly  polarized  sphere  -J  a? 

293.  In  the  case  of  any  purely  polarized  distribution  bounded 
by  a  surface    S,   the   volume    and   superficial   densities   are 
accounted  for  by  a  vector  /,  of  components  A,  B,  C,  such  that 
within  S,  p  =  —  Divergence  /,and  on  $,  <r  =  I-  cos  (n,  7).    Show 
that  the  polarization  might  be  equally  well  represented  by 
any  vector  which  differs  from  /  by  a  solenoidal  vector  O  every 
line  of  which,  if  it  meets  $  at  all,  lies  wholly  on  S.     Is  the 
induction  within  a  hard  magnet  definite  ? 

294.  Matter  is   distributed  on  the  ends  of  a  cylinder  of 
revolution   of   length  I  and   radius  a.     The   density  within 
the  cylinder  and  the  superficial  density  on  its  curved  surface 
are  everywhere  equal  to  zero.     On  one  end  a  quantity  2  TT<I* 
of  matter  is  distributed  with  density  o-  =  a2  /r,  where  r  is  the 
distance  from  the  axis;  on  the  other  end  a  quantity  —  2-n-as 
is  distributed  with  density  a-  =  —  3  r.     Can  you  affirm  that 
the  cylinder  is  not  polarized  solenoidally  ? 

295.  Show  that  if,  in  the  case  of  a  polarization  symmetrical 
about  the  axis  of  z  so  that  the  lines  of  the  vector  /  lie  in  planes 
which  pass  through  this  -axis,  Z  be  the  component  of  I  parallel 
to  the  axis  of  z  and  Mthe  component  perpendicular  to  the 
axis,  p  =  —  [DrR  -f  R/r  +  DZZ~\.    Consider  the  volume  den 
sity  in,  and  the  superficial  density  on,  a  cylinder  of  revolution 
of  length  I  and  radius  a,  the  axis  of  which  coincides  with  the 
axis  of  z,  when  R  =  (r  —  a)f(z),  and 

+  (2  r  -  a)f(z)  /r\  dz. 


f 


MISCELLANEOUS    PROBLEMS. 


449 


Assuming  both  /  and  \j/  at  pleasure,  draw  the  lines  of  polari 
zation  for  the  simple  case  which  you  have  chosen. 

296.  A  solenoidal   vector,  the  components  of  which  par 
allel  to  the  columnar  coordinates  r,  0,  x  are  (a  —  r)f'(x),  0, 
(2  r  —  a)f(x)/r,  represents  the  polarization  within  a  magnet 
bounded  by  the  cylindrical  surface  r  =  a  and  the  planes  x  =  0, 
x  —  b.     Determine  the  surface  density  a-  and  draw  two  of  the 
lines  of  polarization  when  f(x)  =  x.     Show  that  a-  is  zero 
when/(z)  =  sin(Trx/b). 

297.  Show  that  a  vector  the  components  of  which  in  the 
directions  of  the  columnar  coordinates  r,  6,  x  are 


[/'(x)  .  J?(r)],  0,  -/(x)  [I"(r)  +F(r)/r-\, 

is  solenoidal,  and  use  this  form  to  determine  two  or  three 
different  polarizations  within  a  bar  magnet  for  which  both 
p  and  <r  shall  be  everywhere  zero. 

298.  Prove  that  an  infinitely  long  cylinder  of  revolution  of 
radius  a,  the  axis  of  which  coincides  with  the  z  axis,  when 
polarized  uniformly  in  the  direc 
tion  of  the  x  axis,  gives  rise  to  the 
potential  function  —  2  irla^x/r2  at 
outside  points.  This  is  identical 
with  the  potential  function  due 
to  a  plane  doublet  of  strength 
2  irla2  at  the  origin.  Within  the 
cylinder  the  resultant  force  has 
the  intensity  2  7r/and  the  direction 
of  the  negative  x  axis,  while  the 
induction  has  the  intensity  2irl 
and  the  direction  of  the  positive 
x  axis.  The  lines  of  induction 

and  the  lines  of  force  have  the  same  direction  without  the 
cylinder  and  opposite  directions  within.  The  lines  of  force 
are  shown  in  Fig.  133.  Show  that  the  normal  component  of 
the  induction  is  continuous  at  the  surface  of  the  cylinder. 


FIG.  133. 


450  MISCELLANEOUS   PROBLEMS. 

299.  A  solenoidally  polarized  distribution  inside  which  the 
lines  of  polarization  are  straight  and  parallel  need  not  be 
uniformly  polarized. 

300.  Prove  that  the  mutual  potential  energy  of  any  two 
small  magnets  at  a  distance  apart  large  compared  with  their 
linear  dimensions  is 

M-i  -  M2  (cos  <£  —  3  cos  Ol  •  cos  02)  /  ^ 

where  Mly  M2  are  the  moments  of  the  magnets,  <£  the  angle 
between  their  directions,  and  Oly  02  the  angles  which  these 
directions  make  with  a  line  drawn  from  the  centre  of  the 
first  to  the  centre  of  the  second. 

301.  Show  that  for  a  simple  magnetic  shell  in  the  form  of  a 
circle,  the  direction  of  the  vector  potential  at  any  point  is  per 
pendicular  to  a  plane  through  the  point  and  a  normal  to  the 
plane  of  the  shell  through  the  centre.     [St.  Peter's  College.] 

302.  Prove  that  if  m  is  the  pole  strength  of  a  slender, 
straight,  uniformly  magnetized  magnet  AB,  a  vector  poten 
tial   may   be   found   which   has  at   any  point  P  the   value 

—  (cos  PAB  4-  cos  PBA),  where  p  is  the  length  of  the  per 
pendicular  dropped  from  P  on  AH,  produced  if  necessary. 
Show  that  the  direction  of  this  vector  potential  is  perpen 
dicular  to  the  plane  PAB.     [M.  T.] 

303.  Show  that  if  V  is  the  value  of  the  potential  function, 
and  F  that  of  the  vertical  component  of  the  magnetic  force 
at  the  earth's  surface,  the  earth's  field  in  outside  space  may 
be   considered  as  due  to  a  surface  distribution  of  density 

—  F/2  TT  —  F/4  ?ra,  where  a  is  the  earth's  radius. 

304.  A  magnetic  needle  is  placed  near  an  infinite  plane  face 
of  a  mass  of  soft  iron.     Show  that  the  reaction  of  the  iron  on 
the  needle  may  be  represented  as  due  to  a  negative  image  of 
the  needle  in  the  plane  face,  reduced  in  intensity  in  the  ratio 
of  (/x  —  1)  /  (fi  +  1),  where  /x  is  the  permeability  of  the  iron. 
[St.  John's  College.] 


MISCELLANEOUS    PROBLEMS.  451 

305.  A  magnetic  element  /SaVof  pole  strength  a  and  moment  b 
lies  in  a  magnetic  field  which  has  the  potential  function  V. 
Show  that  if  I,  m,  n  are  the  direction  cosines  of  the  axis  of 
the  element,  the  mutual  potential  energy  of  the  element  and 
the  field  is  \E=a(Vy-V^  =  b(l-DxV+m.DyV+n-DsV). 
If  the  element  is  a  rectangular  parallelepiped,  dx  dy  dz,  taken 
from  a  magnetized  body  in  which  the  polarization  is  /, 
b  =  Idx  dy  dz,  &E=  (A-DxV+B-DyV+C.DzV)dxdy  dz, 
and  the  mutual  energy  of  the  field  and  the  magnet  is  the 
integral  of  this  last  expression.  .  If  the  magnet  is  a  simple 
shell  of  strength  4>, 


E  =  *(l  -DtV+m-  Dfl  V+  n  •  Dz  V)  dS,  or 

cos  (*»  n)  +  Y>  cos  0,  n)  +  Z-  cos  (z,  n)']  dS, 


where  the  integration  is  to  be  extended  over  one  face  of  the 
shell. 

306.  A  simple  plane  circular  magnetic  shell  of  radius  r  lies 
in  the  yz  plane,  with  its  centre  at  the  origin,  in  a  magnetic  field 
symmetrical  about  the  x  axis.     The  intensity  of  the  x  compo 
nent  of  the  field  is  F  (x),  where  F  is  a  continuous  function, 
such  that  F  (oo)  =  0.     Show  that  the  force  which  urges  the 
shell  is  equal  to  ira2&  -  DXF.     The  centre  of  the  rigid  shell  is  to 
move  along  the  x  axis  to  infinity  while  the  plane  of  the  shell 
is  parallel  to  the  yz  plane.     Compute  the  work  done  on  the 
shell  by  the  field  during  the  motion.     Has  the  field  any  com 
ponent  perpendicular  to  the  x  axis  ?     Compute  the  work  done 
on  the  shell  by  the  field,  with  the  help  of  the  method  discussed 
at  the  top  of  page  218.     Show  that  a  vector  which  has  the  com- 
ponents  F(x),  y  C/(^+z2)  -  \y  •  F'  (x),  zC/(y>  +  z2)  -  \z  .  F'  (x), 
is  solenoidal  and  is  symmetrical  about  the  x  axis. 

307.  Show  that  if  A1,  B\  C'  are  the  components  of  magneti 
zation  at  the  point  (x',  y\  z')  in  any  magnet,  M',  and  if  p. 
denotes  the  reciprocal  of  the  distance  between  (x',  y',  z')  and 


452  MISCELLANEOUS    PROBLEMS. 

(x,  y,  z),  the  components  at  (a*,  y,  z)  of  the  ordinary  vector 
potential  of  the  magnetic  induction  are 


fff 


(C'.Dx.p-A'.Ds,p)dr', 

-  D,p  -  &  •  DX.P)  dr'. 

The  scalar  potential  function  of  magnetic  force  is  in  the 
same  notation 

+  B-.  !)„!>  +  C'  •  D,p)  dr'. 

If  M'  is  a  simple  shell  of  strength  4>',  the  x  component  of 
the  vector  potential  function  can  be  written  in  the  form 

<!>'  C  C[Djp  •  cos  (y,  n)  —  Dy,p  •  cos  (z,  ri)~\dS'.     [Maxwell.] 

308.  Show  that  if  r  is  the  distance  from  a  fixed  point,  the 
line  integral  around  any  closed  curve  s  of  the  tangential 
component  of  the  vector  (1/r,  0,  0)  is  equal  to  the  surface 
integral,  taken  over  any  cap  S  bounded  by  s,  of 

Dz  (1  IT)  •  cos  (y,  n)  -Dy(l/r).  cos  (z,  n), 

where  n  is  a  positive  normal  to  the  cap.  Obtain  two  similar 
equations  with  the  help  of  the  vectors  (0,  1/r,  0),  (0,  0,  1/r), 
and  prove  that  the  components  of  the  vector  potential  function 
of  the  force  due  to  a  magnetic  shell  of  strength  <I>  in  air  are 

4>  f[cos(x,  s)  •  /r]d8,         4>  J[cos  (y,  s)  •  /r]ds, 
*J  [cog  (a,  *)•/*"]  d*j 

taken  around  the  perimeter  of  the  shell. 

If  (L,  M,  N)  are  the  curl  components  of  a  vector  (Fx,  Fy,  Fz), 
the  latter  is  a  vector  potential  function  of  the  magnetic  field. 


MISCELLANEOUS    PROBLEMS.  453 

Show  that  the  mutual  potential  energy  of  a  magnetic  shell  of 
strength  <!>'  and  the  field  is  -  4>'  C(FM  •  dx'  +  Fy  •  dy'  +  Fz  •  dz'), 

taken  around  the  shell  in  positive  direction.     If  the  external 
field  is  caused  by  another  shell  of  strength  <1>,  we  have 


where  the  integrals  are  to  be  taken  around  the  perimeter  s  of 
the  second  shell,  and  the  mutual  potential  energy  of  the  two 
shells  is 

—  Sx&'J  J  [cos  (x,  s)  •  cos  (x,  s')  +  cos  (y,  *)  -  cos  (y,  5') 

+  COS  (S,  A')  •  COS  (z,  «')]  [flfe  •(/*'/  /•] 

or  -  <M>'  f  f[cos  (C/A-,  ds')  /  r]  rf«  •  ds'. 

The  integral  by  which  —  <M>'  is  multiplied  has  been  called 
the  "  geometric  potential  "  of  the  two  curves. 

309.  Prove  in  two  different  ways  that  the  energy  of  the 
surface  distribution  a-  =  /-cos(?i,  /),  on  a  sphere  of  radius  a 
uniformly  polarized  to  intensity  /,  is   87r2/2a8/9.      In  what 
sense  is  this  the  energy  of  the  distribution?     Give  a  sum 
mary  of  the  reasoning  of  Lord  Kelvin  in  his  paper  "  On  the 
Mechanical  Values  of  Magnets." 

310.  If  a  polarized  distribution  is  placed  in  a  field  of  force 
which  has  a  potential  function  I',  the  mutual  potential  energy 
of  the  field  and  the  distribution  as  a  whole  is 


VI  -  cos  (n,  /)  dS  -  J  J  J    F(/V1  +  />y#  +  Dz  C)  dr, 

where  the  first  integral  is  to  be  extended  over  the  surface  of 
the  distribution  and  the  second  through  its  volume.  Show 
that  this  energy  is  equivalent  to 

(A-DXV+B-DVV+  C-l 


454  MISCELLANEOUS   PROBLEMS. 

311.  A  sphere  of  radius  a  uniformly  polarized  to  intensity 
/  is  placed  in  a  uniform  field  of  force  of  intensity  X.     Show 
that  if  the  directions  of  the  field  and  the  polarization  coincide, 
the  mutual  potential  energy  of  the  sphere  and  the  field  is 
—  47ra3/X/3.     What  would  be  the  energy  if  the  direction 
of  the  field  and  polarization  were  opposed  ?     It  would  be  zero 
if  these  directions  were  perpendicular  to  each  other. 

312.  If  V  is  the  potential  function  due  to  a  volume  distri 
bution  of  density  pl  in  a  region  Tl9  and  a  surface  distribution 
of  density  <TI  on  a  surface  Si,  and  if  U  is  a  continuous  function 


+  4 


CC  Ucr, 


where  the  volume  and  surface  integrations  in  the  second 
member  are  to  be  extended  respectively  through  and  over  a 
spherical  surface  of  radius  r  so  large  as  to  include  2\  and  S^ 
If  U  vanishes  at  infinity,  the  last  surface  integral  vanishes 
when  r  is  infinite.  Use  this  equation  to  compute  the  mutual 
potential  energy  (—  £  ira*IX)  of  a  sphere  of  radius  a,  uni 
formly  polarized  to  intensity  /  in  the  direction  of  the  x  axis, 
and  a  uniform  field  (X,  0,  0),  in  which  it  lies.  In  this  case  the 
value  of  the  last  term  in  the  second  member  is  —  32  ?r2a8/X/9. 

313.  At  a  distance  of  10  centimetres  from  the  middle  point 
of  a  wire  140  centimetres  long,  the  magnetic  force  due  to  a 
current  in  the  wire  would  be  within  one  per  cent  of  that 
which  would  be  produced  if  the  wire  were  infinite. 

314.  Show  that  the  magnetic  force  within  a  square  circuit 
(of  side  =  2a)  at  a  point  midway  between  two  sides,  at  a  dis 
tance  x  from  the  centre  of  the  square,  is 


a  \ 


a  —  x  a  +  x 


MISCELLANEOUS    PROBLEMS.  455 

Draw  a  curve  which  shall  represent  this  force  as  a  function 
of  x.  What  if  x  is  greater  than  a  ?  Show  that  the  force  at 
a  point  distant  y  from  the  plane  of  the  circuit  in  the  axis  of 

the  circuit  is 

SCa2  1          } 

j 


Show  that  if  a  current  of  A  amperes  be  sent  through  a 
tangent  galvanometer  which  has  a  square  coil  consisting  of 

n  turns  of  wire, 

5  all  tan  8 
A  —  -  -p  -- 
2V2-7* 

315.  If  a  circuit  carrying  a  steady  current  C  is  a  regular 
polygon  of  2  n  sides,  and  if  a  is  the  radius  of  the  inscribed  circle, 
the  magnetic  force  at  the  centre  is  (4  n  C/a)  -  sin(7r/2  n). 

316.  The  plane  of  the  ring  of  a  tangent  galvanometer  which 
consists  of  a  single  turn  of  fine  wire  is  the  vertical  plane  of 
the  magnetic  meridian.     Show  that  if  a  current  of  A  amperes 
be  sent  through  the  ring,  the  strength  of  the  field  at  a  point 


P  in  its  axis  at  a  distance  z  from  the  centre  is 

»(*  - 

where  r  is  the  radius  of  the  ring.  Hence  prove  that  if  the 
centre  of  the  galvanometer  needle  is  at  P,  the  deflection  will 
be  given  by  the  equation  A  =  [  5  (z2  -f  r2)3  IT  tan  a]  /in*. 

317.  Show  that  at  a  point  on  the  axis,  at  a  short  distance 
(z)  from  the  centre  of  a  tangent  galvanometer  coil  of  radius 
a,  the  intensity  of  the  electromagnetic  field  due  to  a  steady 
current  passing  through  the  coil  is  to  the  intensity  of  the 

same  field  at  the  centre  asfl-       ^  )  to  1,  nearly. 


318.  Show  that  if  around  a  ring  formed  of  a  piece  2  b  centi 
metres  long  of  a  thin  metal  tube  of  inside  radius  a  and  of 
outside  radius  a  +  8,  a  steady  current  of  strength  2b8C  uni 
formly  distributed  through  the  conductor  could  be  sent,  the 


456  MISCELLANEOUS    PROBLEMS. 

strength  of  the  resulting  electromagnetic  field  at  the  centre 

A         S~il  £ 

of  the  axis  of  the  coil  would  be  — =  =.     What  would  this 

intensity  become  if  the  tube  were  to  shrink  indefinitely  in 
length  while  the  whole  current  around  it  remained  unchanged  ? 
Assuming  that  when  x  is  small 

(1  +  x)-*  =  1  -  1  x  +  l&  -  ^  .r>  -f  .  •  • , 
deduce  from  your  results  the  usual  correction,  —  — — ,  for  the 

breadth  of  the  coil  of  a  tangent  galvanometer. 

319.  The  vertical  coil  of  a  tangent  galvanometer  makes  a 
small  angle  8  with  the  east  and  west  line  through  the  centre 
of  its  needle.     If  a  steady  current,  of  such  strength  that  it 
would  cause  a  deflection  of  45°  if  the  plane  of  the  coil  were 
in  the  meridian,  be  now  sent  through  the  coil,  it  will  cause  a 
deflection  of  £8. 

320.  The  centres  of  the  rings  of  a  two-coil  tangent  galva 
nometer  are  20  centimetres  apart  and  the  mean  radius  of  each 
of  the  coils   is   20    centimetres.     The  centre  of  the  needle 
is  on  the  common  axis  of  the  coils  halfway  between  their 
centres.     When  the  instrument  is  properly  set  up  in  a  cer 
tain  place  a  steady  current  of  half  an  ampere  sent  through 
both  coils  in  series  causes  a  deflection  of  45°.     Show  that  if 
there  are  20  turns  in  each  coil,  ff=  167T/10  (5)3. 

321.  A  tangent  galvanometer  has  two  equal  vertical  coils,  each 
of  mean  radius  r,  placed  at  a  distance  apart  of  2  r  ( VS  —  1)*. 
The  short  compass  needle  is  placed  midway  between  the  coils 
on  their  common  axis.     Show  that  the  needle  deflection  caused 
by  any  current  which  passes  in  the  same  direction  through 
both  coils  in  series  will  be  the  same  as  if  the  same  current 
passed  through  only  one  coil,  while  the  centre  of  the  needle 
was  at  the  centre  of  this  coil. 

322.  Show  that  if  the  vertical  coil  of  a  tangent  galva 
nometer  makes  an  angle  6  with  the  meridian,  and  if  a  current 


MISCELLANEOUS    PROBLEMS.  457 

of  C  amperes  be  sent  through  it  first  in  one  direction  and  then 
iii  the  other,  causing  deflections  of  ^  and  S2  respectively,  then 

•n-nC  _        sinSi  sin82 

5777  ~~  cos  (0  -  ^  ~  cos  (0  +  82)' 

and  tan  0  =  £(ctn  &,  —  ctn  8^,  where  r  is  the  mean  radius  of 
the  coil  and  n  the  number  of  turns  of  wire  on  it. 

323.  From  a  thin,  flat  sheet  of  copper  of  thickness  8  is  cut 
a  ring  of  inside  radius  a  —  d  and  outside  radius  a  —  d.  If  a 
steady  current  of  strength  2C&d  could  be  made  to  circulate 
around  this  ring,  what  would  be  the  strength  of  the  electro 
magnetic  field  at  the  centre  of  the  ring?  What  would  this 
strength  become  if  the  ring  were  to  shrink  to  a  fine  wire  ring 
of  radius  a  concentric  with  the  original  ring  without  change 
of  the  current  strength  ?  Assuming  that  when  x  is  small 


deduce  from  your  results  the  usual  correction  (one-twelfth  of 
the  square  of  the  ratio  of  the  depth  of  the  coil  to  its  mean 
radius)  for  the  depth  of  the  ring  of  a  tangent  galvanometer. 

324.  A  certain  galvanometer  coil  is  wound   upon  a  large 
square  frame.     When  the  vertical  plane  of  the  coil  makes  an 
angle  0  with  the  meridian  a  certain  current  C  sent  through 
the  coil  deflects  the  short  needle  through  an  angle  £  0  towards 
the  coil.     Show  that  C  would  cause  the  same  deflection  if  the 
coil  were  in  the  meridian. 

325.  On  the  axis  of  a  fixed  circular  ring  of  wire  which  car 
ries  a  steady  current  C  is  a  molecular  magnet  of  moment  m. 
Show  that  if  the  axis  of  the  magnet  makes  an  angle  0  with 
the  axis  of  the  ring  the  moment  of  the  couple  which  tends  to 
diminish  0  is  (2  irmC  sin8  <£  •  sin  6)  /a,  where  a  is  the  length  of  a 
radius  of  the  ring  and  <£  the  angle  subtended  at  the  molecule 
by  the  radius. 


458  MISCELLANEOUS    PROBLEMS. 

326.  A  perfectly  flexible  wire  fastened  at  two  fixed  points 
carries  a  current  of  given  strength.     Prove  that  in  a  uniform 
field  of  magnetic  force  it  will  tend  to  assume  the  shape  of  a 
helix.     [M.  T.] 

327.  A  very  long  straight  wire  which  carries  a  steady  cur 
rent  C  is  at  right  angles  to  the  plane  of  a  circular  ring  of 
radius  a  which  carries  a  current  C'.     The  ring  is  free  to  turn 
about  the  diameter  which  intersects  the  straight  wire.     Prove 
that  the  couple  tending  to  turn  the  ring   is  2TrCC'a2/r  or 
27r(7C"r,  according  as  a  is  less  or  greater  than  ?•,  the  distance 
of  the  wire  from  the  centre  of  the  ring.     [Trinity  College.] 

328.  A  plane  ring  can  move  about  a  diameter  parallel  to  an 
infinite  straight  wire,  the  distance  of  which  from  the  centre 
of  the  ring  is  equal  to  the  radius  of  the  latter.     Show  that 
when    currents   CC'  are    sent   through  the   two  circuits  the 
couple  tending  to  turn  the  ring  is 

4  7rCC'a(cos  </>  -  cosi  <£/  V2  cos  0), 

when  a  is  the  radius  of  the  ring  and  <j>  the  acute  angle  which 
the  normal  to  its  plane  makes  with  the  perpendicular  to  the 
straight  wire  drawn  from  the  centre.  [M.  T.] 

329.  If  a  layer  of  n'  turns  of  wire  carrying  a  steady  current 
of  unit  strength  and  forming  a  coil  k'  be  wound  uniformly  on 
such  a  ring  coil,  k,  as  that  shown  in  Fig.  77,  the  induction  due 
to  the  current  in  k'  has  at  every  point  within  the  coil  the  value 
2/xw'/r.     The  integral  of  n  times  this  quantity  taken  over  a 
cross-section  of  the  ring  R  on  which  k  is  wound  gives  the 
mutual  inductance  of  the  two  coils.     Show  that  if  R  may  be 
regarded  as  formed  by  revolving  a  circle  of  radius  a  about  a  line 
in  its  plane,  distant  b  from  its  centre,  the  value  of  the  integral 
is  4  TTfjinn'  (b  —  V&2  —  a2).     If  R  were  formed  by  the  revolution 
of  a  rectangle  with  sides  of  length  b  parallel  to  the  axis  and  a 
perpendicular  to  it,  the  value  of  the  integral  would  be 

7  ,      c  4-  a  /2 
2  nn 'fib  log 


MISCELLANEOUS    PROBLEMS.  459 

where  c  is  the  distance  of  the  centre  of  the  section  of  the 
ring  from  the  axis.  Show  that  the  self-inductance  of  k 
might  be  found  for  the  two  cases  just  mentioned  by  putting 
n  equal  to  ri  in  the  expressions  for  mutual  inductance,  and 
imagining  k'  to  move  into  coincidence  with  k. 

330.  A  thin  tubular  conductor  of  circular  section  has  a 
radius  a  and  carries  a  steady  current    (7;    prove   that   the 
mechanical    action   between    the   different    portions   of    the 
current  produces  a  transverse  tension  in  the  tube,  of  intensity 
C*/7ra.     [St.  John's  College.] 

331.  The  ponderomotive  forces  which  act  upon  a  portion 
A^A2  of  a  circuit  which  carries  a  steady  current  of  strength 
C  in  the  field  of  a  magnetic  pole  of  strength  m  at  the  point  0, 
have  a  resultant  moment  M  about  any  straight  line  OZ  drawn 
through  0.     Let  PP'  represent  an  element  As  of  the  circuit ; 
let  OP  =  r,  OP'  =  r  +  Ar,  ZOP  =  0,  ZOP'  =  0  +  A0,  (/•,*)  =  8, 
and  denote  the  angle  between  the  planes  ZOP  and  POP'  by  <£. 
The  fundamental  equation  of  spherical  trigonometry  yields 

cos  (0  +  A0)  =  cos  0  •  cos  POP'  +  sin  0  •  sin  POP'  •  cos  <f>, 

and  it  is  evident,  since  A0  is  not  greater  than  POP',  that  the 
limit  of  the  ratio  of  (cos  A0  -  cos  POP1)  /sin  POP'  is  zero, 
so  that  cos<£  is  approximately  equal  to  —  sin  A^/sin  POP'. 
The  Theorem  of  Sines  applied  to  the  plane  triangle  POP' 
yields  the  equation  PP' /  OP'  =  sin  POP'/sin  OPP'.  Prove 
that  the  moment  about  OZ  of  the  elementary  force  exerted 
by  the  pole  upon  As  may  be  written, 

AJf  =  mC-  sin  8  •  cos  <£  •  sin  0  •  As/r, 

and  that  for  purposes  of  integration  this  is  equivalent 
to  —  mC •  sin  0  •  dO,  so  that  M  =  mC(cos  02  —  cos  ^),  where 
0!  =  ZOAlt  62  =  ZOA2.  If  Ol  =  02,  as  in  the  case  of  a  closed 
circuit,  M  is  zero.  Consider  the  possibility  of  rotation  about 
a  straight  line,  of  a  closed  circuit  bearing  a  steady  current  C 
under  the  action  of  any  number  of  magnetic  poles  on  the  line. 


460  MISCELLANEOUS    PROBLEMS. 

332.  In  the  case  of  a  solitary  linear  circuit  s  carrying  a 
current  C  in  its  own  magnetic  field,  all  the  lines  of  force  are 
closed  curves  threading  the  current,  and  the  line  integral  of 
the  force  taken  around  any  one  of  these  curves  is  4?rC.  The 
equipotential  surfaces  fill  all  space  ;  each  of  them  is  a  cap 
bounded  by  the  circuit,  and  the  surface  integrals  of  the  induc 
tion  taken  over  these  caps  are  all  equal.  Use  the  reasoning 
of  page  270  to  show  that  since 


=  Cff-ds,  and^=   C  C 


-  dBdr, 


where  dp  is  the  increment  of  the  induction  flux  through  the 
circuit,  due  to  a  small  increase  in  the  current.  'Show  from  the 
equation  E  —  dp/dt  =  rC  that,  besides  the  energy  dissipated 
in  heat,  the  generator  in  a  solitary  circuit  must  furnish  an 
amount  of  energy  C  •  dp  while  the  current  in  the  circuit  is 
changed  from  C  to  C  +  dC,  and  that  the  difference  dW 
between  this  quantity  and  the  increment  dT  of  the  electro- 
kinetic  energy  shows  the  amount  of  energy  which  is  used  in 
some  other  way  than  in  increasing  this  energy. 
Prove  that 


and  use  this  expression  to  compute  (see  page  291)  the  energy 
loss  due  to  hysteresis  during  a  cycle  of  magnetization. 

333.  The  distance  between  the  axes  of  two  infinitely  long, 
straight,  round,  non-magnetic  wires  (Alf  A2)  of  radius  a  and 
parallel  to  each  other  is  b.  One  wire  carries  a  steady  current 
(7,  uniformly  distributed,  in  one  direction,  and  the  other  wire 
an  equal  current,  uniformly  distributed,  in  the  other  direction. 


MISCELLANEOUS    PROBLEMS.  461 

If  the  cross-section  of  each  wire  be  divided  into  n  elements  of 
equal  area,  every  element,  dS,  is  the  section  of  a  filament  which 
carries  a  current  CdS/wa2.  Imagine  a  circuit  made  up  of  a 
certain  filament  Fl  in  A^  distant  r,',  r^'  from  the  axes  of  the 
wires,  and  a  filament  F.2  in  A.,  distant  r2',  r2"  from  these  axes. 
The  flow  of  induction  through  this  circuit  due  to  A^  is 

/»«  9  Cr  Cri  2  C 

I          -dr  +  I    '     -dr  or  2  C[(a2  -  r/2)/2  a2  +  log  (r2'/a)] 

*/r'        CL  \s  a  T 


and  that  due  to  As  is  2C[(a2  -  r2"2)/2a2  +  log  «/«)]•  If 
the  filaments  are  symmetrically  situated,  >-2'  =  r/',  r/  =  r2"  and 
the  induction  through  the  circuit  is 


The  electrokinetic  energy  of  a  set  of  circuits  is  equal  to 
one-half  the  sum  of  the  products  formed  by  multiplying  the 
induction  through  any  circuit  by  the  current  in  that  circuit. 
The  contribution  which  the  elementary  circuit  just  mentioned 
would  make  to  the  electrokinetic  energy  T  is,  therefore,  one- 
half  the  product  of  the  induction  through  it  and  the  current 
which  it  carries,  so  that 

T=  —.  ;  f  fjl  -  r/'/Sa'  _  r>«/2  a'-  +  log(r,'/a) 

TTO"*/    •/ 


where  the  integration  is  to  be  extended  over  all  such  elementary 
circuits,  that  is,  over  the  cross-section  of  either  wire.  We 
may  write  for  dS,  either  r/  •  drj  •  dOl  or  rz"  •  dr2"  •  dO»  at  pleasure, 
and  we  may  use  the  first  of  these  for  the  first,  second,  and 
fifth  terms  of  the  integrand,  and  the  second  form  for  the  other 
terms.  The  limits  of  0  will  be  0  and  2  TT,  and  those  of  >•/  and 
r/',  0  and  a.  Assuming  that,  if  m>n, 


\     log  (m  -f  n  cos  0)  dd  =  TT  log  \  J  (i 


462  MISCELLANEOUS    PKOBLEMS. 

show  that  T=  <72[i  +  log(62/a2)]  and  that  it  makes  no  differ 
ence  how  the  separate  filaments  of  Al  and  A2  are  combined 
into  elementary  circuits.  Show  also  that  the  inductance,  per 
unit  of  its  length,  of  the  circuit  made  up  of  the  wires,  is 
2  log  (b2/ a2)  -f  1.  Prove  that  if  the  wires  had  inductivities 
/AL  and  /u,2  and  radii  aly  a2,  we  should  have 

L  =  2p  log/ (&Xa2)  +£(/*!  +  /x2), 

where  /u,  is  the  inductivity  of  the  surrounding  medium.  [For 
a  discussion  of  the  inductance  of  the  circuit  when  the  cross- 
sections  of  the  long  parallel  wires  are  of  any  form,  the  reader 
is  referred  to  A.  Gray's  Absolute  Measurements  in  Electricity 
and  Magnetism,  Vol.  II,  p.  288,  and  to  Drude's  Physik  des 
Aethers,  p.  207.] 

334.  Obtain    Heaviside's    expressions    (Electrical   Papers, 
p.  101)  for  the  coefficients  L19  L2,  M,   of  self  and  mutual 
induction  for  two  parallel  wires  of  length  I,  radii  aly  «2,  and 
inductivities  /x1?  ^  suspended  parallel  to  each  other  and  to 
the  earth  at  heights  hl}  h2  and  at  a  horizontal  distance  d 
apart,  if  the  current  is  supposed  to  return  through  the  earth 
in  a  thin  sheet,  and  if  h±  and  A2  are  small  compared  with  L 
These  expressions  are 

L,/l  =  J^  +  2  log  (2  /h/fli),  L2/l  =1/^  +  2  log (2  h2/a2), 

™  n        i         <&  +  (^1  +  hzY 
M/l  =  log  — Yf 7^  • 

335.  Show  that  if  K  and  //,  represent  the  dimensions  of 
electric  and  magnetic  inductivities  respectively,  the  dimen 
sions  of  /ut  in  terms  of  L,  M,  T,  K  are  L~2T2K-1,  while  those 
of  K  in  terms  of  L,  M,  T,  /x  are  L~2T2^-1.     Show  that  the 
dimensions    of    electric    quantity    in    the    two    systems    are 

f*,  Zi*Jf*fi   *j  those  of  magnetic  quantity 
.* ;  those  of  electric  field  strength  L 
* ;    those  of  magnetic  field  strength 


MISCELLANEOUS   PROBLEMS.  463 

i  those  of  electric  potential  LrM*T~lK~h, 
those  of  magnetic  potential  L*M*T~'2K*, 
tnose  of  conductivity  LT~1K,  Z"1!/*"1;  those 
of  electric  current  L*M-T~'2K',  L-M*T~  V~"  5  those  of 
capacity  LK,  L~1T2P~1;  those  of  inductance  L~1T'K~\  Z/x; 
those  of  magnetic  moment  L*M*K~*,  l£lf*2r"V*;  those  of 
electric  surface  density  L~*M*T~*K*,  L~*M*n~*- 

336.  A  rigid  plane  wire  of  any  shape  is  free  to  turn  about 
a  point  0  in  its  plane  distant  a  and  b  from  the  nearer  and 
farther  ends  of  the  wire.     The  plane  of  rotation  is  perpen 
dicular  to  the  lines  of  a  uniform  field  of  induction  of  intensity 
B.     Show  that  if  the  wire  forms  part  of  a  circuit  which 
carries  a  current  (7,  the  moment  about  0  of  the  forces  which, 
acting  on  the  wire  tend  to  set  it  in  motion,  is  ^BC(b2  —  a-). 
If  the  wire  rotates  with  angular  velocity  to,  it  cuts  the  lines 
of  the  field  at  the  rate  ^  o>B  (lr  —  a2). 

337.  A  copper  disc  perpendicular  to  the  lines  of  a  uni 
form  magnetic  field  is  spun  in  its  own  plane  about  a  fixed 
point  0  and  is  continuously  touched  at  two  points  by  the 
fixed  electrodes  of  a  galvanometer.     Show  that  the  current 
in  the  galvanometer  is  proportional  to  the  difference  of  areas 
swept  out  by  the  radii  vectores  from  O  to  the  points  touched. 
[M.  T.] 

338.  A  magneto-electric  machine,  driven  at  a  constant  rate, 
sends  current  through  the  coil  of  another   magneto-electric 
machine  used  as  a  motor.     When  the  second  machine  is  held 
still,  a  power  JFis  used  in  the  circuit.    Prove  that  the  maximum 
power  obtainable  from  the  second  machine  is  J  JF,  and  that 
then  the  first  machine  absorbs  ^  W  from  the  engine  which 
drives  it.     [M.  T.] 

339.  Show  that  (1)  if  a  conductor  be  moved  along  a  line  of 
magnetic  induction   parallel  to  itself,  it  will   experience  no 
electromotive  force ;  (2)  if  a  conductor  carrying  a  current  be 
free  to  move  along  a  line  of  magnetic  induction,  it  will  experi 
ence  no  tendency  to  do  so ;  (3)  if  a  linear  conductor  coincide 


464  MISCELLANEOUS    PROBLEMS. 

in  direction  with  a  line  of  magnetic  induction  and  be  moved 
parallel  to  itself  in  any  direction,  it  will  experience  no  electro 
motive  force  in  the  direction  of  its  length ;  (4)  if  a  linear 
conductor  carrying  an  electric  current  coincide  in  direction 
with  a  line  of  magnetic,  induction,  it  will  not  experience  any 
mechanical  force.  [Maxwell.] 

340.  Discuss  the  following  statements  of  different  writers  : 
"When  an  electromotive  force  E  is  suddenly  applied  to  an 
inductive  circuit  of  resistance  J?,  the  counter-electromotive 
force  of  self-induction  is  initially  equal  to  E  and  the  current 
caused  by  E  is  initially  zero."     '•  When  the  current  is  rising 
a  portion  of  E,  viz.,  .#(?,  is  employed  in  maintaining  according 
to  Ohm's  Law  the  current  C  already  established ;  the  other 
portion  of  E,  viz.,  L  •  DtC,  is  employed  in  increasing  the  electro 
magnetic  momentum  LC."     "At  the  beginning  the  whole  of 
the  electromotive  force  acts  to  increase  the  current."     "  If  a 
current  is  established  in  a  coil  and  the  coil  left  to  itself,  short 
circuited  without    any  electromotive    force  to  maintain   the 
current,  then  as  the  decaying  current  reaches  a  value  C  the 
electromotive  force  RC  is  equal  to  —  L '  DtC" 

"The  reactance  does  not  represent  the  expenditure  of 
power,  as  does  the  effective  resistance,  r,  but  merely  the 
surging  to  and  fro  of  energy.  While  the  effective  resistance, 
r,  refers  to  the  energy  component  of  the  applied  electromotive 
force  or  the  electromotive  force  in  phase  with  the  current,  the 
reactance,  x,  refers  to  the  wattless  component  of  the  electro 
motive  force  or  the  electromotive  force  in  quadrature  with 
the  current." 

341.  Compare  the  differential  equation  of  motion  of  a  body 
of  mass  L,  moving  with  velocity  C,  under  the  action  of  an 
impressed  force  E,  which  tends  to  increase  the  velocity,  and 
a  resistance  rC,  proportional  to  the  velocity,  with  the  equa 
tion  which  the  current  in  an  inductive  circuit  must  satisfy. 
Why  should  LC  in  the  electrical  case  be  called  the  "electro 
magnetic  momentum  "  ? 


MISCELLANEOUS    PROBLEMS.  465 

342.  Given  that 

j  sinpt  •  sin(^£  -f-  a)  •  dt 

=  [(2pt  —  sin  2 pt)  cos  a  —  cos  2 pt  -  sin  a]  /4j9, 

show  that  the  average  value  for  any  number  of  whole  periods 
of  the  product  of  two  simple  harmonic  functions  of  the  same 
period  is  half  the  cosine  of  their  phase  difference.  Show  that 
the  activity,  or  "  power/'  of  a  harmonic  alternating  current 
is  equal  to  the  product  of  the  effective  current,  the  effective 
electromotive  force  applied  to  the  circuit,  and  the  cosine  of 
their  difference  of  phase. 

343.  Two  coils,  the  resistances  of  which  are  rly  r2  and  the 
inductances  Lu  L»  are  in  series  in  a  simple  circuit  carrying  a 
harmonic  current.     Is  the  impedance  of  the  two  taken  together 
equal  to  the  sum  of  their  impedances  ? 

344.  The  ends  of  the  coil  of  an  electromagnet  are  subject 
to  a  rapidly  alternating  electromotive  force.     Show  that  the 
energy  expended  in  the  battery  when  a  given  amount  of  heat 
is  produced  in  the  wire  will  be  greater  than  would  be  the 
case  if  the   electromotive   force   were  constant  in   direction 
and  magnitude. 

345.  Show  that  if  a  number  of  linear  circuits  slt  s2,  s3,  •  •  • , 
carrying  currents  C\,  C2,  C3,  •  •  •,  exist  together,  and  if  the  total 
flow  of  induction  through  st  be  denoted  by  j^.,  the  electro- 
kinetic  energy,  T,  may  be  written  $2Ckpk  and  pk  =  %DckT. 
The  quantity  pk  is  sometimes  called  the  electrokinetic  momen 
tum  of  sk.     Compare  this  result  with  the  equation,  pk=DckT, 
given  on  page  296. 

346.  Show  that  the  total  flux  of  electric  current  induced  in 
a  thin  circular  coil  ofc  radius  a  and  resistance  R,  made  up  of 
n  turns  of  wire,  when  the  coil  is  turned  through  two  right 
angles  in  the  earth's  uniform  magnetic  field  H,  is  2  ira?nH / R. 
Show  also  that  if  a  sphere  of  soft  iron  of  the  same  radius  a 
be  pushed  completely  into  the  opening  in  the  coil,  the  flux  is 
increased  in  the  ratio  of  3 /*/(//,  +  2).     [St.  John's  College.] 


470  MISCELLANEOUS    PROBLEMS. 

the  currents  (Il9  /2),  similarly  measured,  in  tlie  two  branches. 
If  one  of  the  branches  (TI)  of  the  di\  ided  circuit  is  non-inductive, 
the  instantaneous  difference  of  potential  between  its  ends  is 
rlCl  and  the  instantaneous  rate  of  expenditure  of  power  in 
the  other  branch  is  r^C^  the  average  activity  in  this  latter 
branch  is  i\  times  the  average  value  of  C1C2.  Show  that  the 
power  expended  in  r.2  is  ^^(J2  — /x2  — /22).  What  is  the 
"  Three  Ammeter  Method  "  of  measuring  power  ? 

363.    Show  that  if  u  -f  vi  is  a  solution  of   the   equation 


L-DtC 


«) 


v  is  a  solution  of  the  equation 

L  -  DtC  +  rC  =  E^  sin  (pt  +  a). 
Prove  that  the  complete  solution  of   this  last  equation  is 


Ae~rt/L  +  E0  •  sin  (pt  +  fl)  /  Vr2  + 

where      tan  b  =  (r  •  sin  a  —  Lp  •  cos  a)/(r-  cos  a  -f-  Lp  -  sin  a). 
364.    Show  that  if  u  and  v  are  solutions  of  the  equations 

L.DtC  +  rC=El-sin  (pj  +  a^, 
^  __  L  •  DC  +  rC  =fi2>  sin  (p2t  +  «2) 

respectively,  u  +  v  is  a  solution  of  the  equation 

L  -  DC  +  rC  =  E-  sin      lt  +  %  +  ^2  •  sin 


and  write  down  the  complete  solution  of  this  last  equation. 
Write  down  also  an  expression  for  the  current  in  an  inductive 
circuit  to  which  n  simple  harmonic  electromotive  forces  of 
given  periods  are  applied. 

365.    Show  that  if  u  +  vi  is  a  solution   of   the   equation 


u  is  a  solution  of  the  equation 

r  •  DtC  +  C/k  =pE0  •  cos  (pt  +  a). 


MISCELLANEOUS   PROBLEMS.  471 

Prove  that  the  complete  solution  of  this  last  equation  is 


Ae-t/kr  +  kpE*  sin  (pt  +  P)  /  VI  + 
where  tan  ft  =  (rpk  sin  a  +  cos  a)  /  (rpk  cos  a  —  sin  a). 

366.    Show  that  if  u  and  v  are  solutions  of  the  equations 
r  •  DtC  +  C/k  =  Al  •  cos  (pj  +  «i), 
r.DtC  +  C/k  =  A«'  cos  (pj  +  a2), 
u  +  -y  will  be  a  solution  of  the  equation 

=Ai>  cos  (^  +  a,)  +  A,  •  cos  (^  +  a2), 


and  write  down  the  complete  solution  of  this  last  equation. 
Write  down  also  an  expression  for  the  current  in  a  non- 
inductive  circuit  of  capacity  k  to  which  n  simple  harmonic 
electromotive  forces  of  given  periods  are  applied. 

367.    If  <£  stands  for  the  operation  Dt,  <£2  for  the  operation 
Dfj  and  so  on,  the  result  of  applying  the  operation 


(po 
to  e**  is  equal  to  the  product  of  ekt  and 


Show  that  if  we  denote  the  result  of  applying  the  operation 


to  e**  by  u,  so  that  u  satisfies  the  equation 

feo  +  Vi<t>  4-  fi'a^8  +  •••) 
a  special  value  of  u  is  the  product  of  ekt  and  the  fraction 

2  4-  • 


Show  how  the  complete  value  of  u  might  be  found  and 
why  the  special  solution  alone  is  needed  in  many  practical 
problems. 


470  MISCELLANEOUS    PROBLEMS. 

the  currents  (I19  /2),  similarly  measured,  in  the  two  branches. 
If  one  of  the  branches  (r^  of  the  divided  circuit  is  non-inductive, 
the  instantaneous  difference  of  potential  between  its  ends  is 
i\Cl  and  the  instantaneous  rate  of  expenditure  of  power  in 
the  other  branch  is  r±  C^  C2 ;  the  average  activity  in  this  latter 
branch  is  i\  times  the  average  value  of  CiC2.  Show  that  the 
power  expended  in  r2  is  %  r±  (P  —  If  —  /22).  What  is  the 
"  Three  Ammeter  Method  "  of  measuring  power  ? 

363.    Show  that  if  u  +  vi  is  a  solution  of   the   equation 


v  is  a  solution  of  the  equation 

L  •  DtC  +  rC  =  E0  -  sin  (pt  +  a). 
Prove  that  the  complete   solution  of   this  last  equation  is 


Ae~rt/L  +  E0  •  sin  (pt  +  ft)  /  Vr2  + 

where      tan  b  =  (r  -  sin  a  —  Lp  •  cos  a)/(r-  cos  a  +  Lp-  sin  a). 
364.    Show  that  if  w  and  v  are  solutions  of  the  equations 

sin  (pj  +  a^, 
sin  (^  +  «2) 

respectively,  M  +  v  is  a  solution  of  the  equation 
L  -  DC  +  rC  =  E-  sin      j  +  a,  +  ^  •  sin 


and  write  down  the  complete  solution  of  this  last  equation. 
Write  down  also  an  expression  for  the  current  in  an  inductive 
circuit  to  which  n  simple  harmonic  electromotive  forces  of 
given  periods  are  applied. 

365.    Show  that  if  u  +  vi  is  a  solution   of   the   equation 


u  is  a  solution  of  the  equation 

r  -  DtC  +  C/lc  =pE0  •  cos  (pt  +  a). 


MISCELLANEOUS   PROBLEMS.  471 

Prove  that  the  complete  solution  of  this  last  equation  is 


Ae-"*  +  kpE0  sin  (pt  +  /?)/  VI  + 
where  tan  /?  =  (rpk  sin  a  -f-  cos  a)  /  (r/>&  cos  a  —  sin  a). 

366.    Show  that  if  u  and  v  are  solutions  of  the  equations 
r  •  DtC  +  C/k  =  AI  •  cos  (p^t  +  «t), 
r'D(C  +  C/k  =  A.,-  cos  QA/  +  «2), 
?*  +  v  will  be  a  solution  of  the  equation 

r  •  DtC  4-  C/&  =  Al  •  cos  (^  +  e^)  +  .42  •  cos  (pj  +  «2)> 


and  write  down  the  complete  solution  of  this  last  equation. 
Write  down  also  an  expression  for  the  current  in  a  non- 
inductive  circuit  of  capacity  k  to  which  n  simple  harmonic 
electromotive  forces  of  given  periods  are  applied. 

367.    If  <£  stands  for  the  operation  Dt,  <f>2  for  the  operation 
Dt2,  and  so  on,  the  result  of  applying  the  operation 


to  e**  is  equal  to  the  product  of  ekt  and 


Show  that  if  we  denote  the  result  of  applying  the  operation 

^  =  (Po  +Pi<t>  +P*tf  +  •  •  -)/fe  +  !/i*  +  <l*f  +  •••) 
to  ef*  by  u,  so  that  u  satisfies  the  equation 

Go  +  ?i*  +  <!<&  +  •••)«  =  (^o  +^i*  -f  ^2*2  +  •  •  •)«**> 
a  special  value  of  M  is  the  product  of  ekt  and  the  fraction 


Show  how  the  complete  value  of  u  might  be  found  and 
why  the  special  solution  alone  is  needed  in  many  practical 
problems. 


472  MISCELLANEOUS    FKOBLEMS. 

Compare  the  results  of  applying  the  operation  Z  and  the 
operation  (A  +  B<f>)  /  (C  +  Zty),  where 

A  =  (PQ  -p2k*  +  ptk*  ----  ),  B=(Pl  -psk2 

C=(q0-  q,k*  +  qtf  ----  ),  D  =  (q,  -  q3k2 
to      M  -sin(A;£  -f  e). 

Show  that  (a  +  b<f>)  •  [M>  sm(kt  +  a)] 


=  M  Va2  +  b2k2  -  sin  [kt  +  a  +  tan-1  (bk/a)~\ 
or  M  -\la?  +  b2k2  -  sin  (A:«  +  8), 

where     tan  8  =  (bk  cos  a  +  a  sin  a)  /  (a  cos  a  —  bk  sin  a), 
and  that  a  special  value  of 


Va2  +  We2      . 
is      M —  •  sin 

Vc2  +  <W 

This  symbolic  notation  is  treated  at  length  in  Forsyth's 
Treatise  on  Differential  Equations  and  in  Perry's  Calculus 
for  Engineers. 

368.    Prove  that  if  <£  stands  for  the  operation  Dt) 

!>2)[Jf.sin(fo 


where  tan  X  =  bk/  (a  —  ck2), 

and  that  a  special  value  of 

m(kt  -f  a)] 


is  Jf  •  sin  (kt  +  a.  —  p)/  V(7  +  rJr)2  +  w2/c2, 

where    tan  ^  =  mk/(l  —  nk2).     Hence    show    that    a   special 
value  of 


.     . 

is  J/     .1  =  •  sm/^  +  a  +  A  - 


MISCELLANEOUS    PROBLEMS.  473 

369.    If  <f>  were  an  algebraic  quantity,  the  expression 

1 


would  be  equivalent  to 

1 


J_/ 

-  «  \ 


Are  these  two  operations  equivalent  when  <£  =  Dt  and  when  a 
special  value  suffices? 

370.  If  an  electromotive  force  E  =  Em  sinpt  be  applied  to 
a  circuit  consisting  of  a  coil  of  resistance  r  and  inductance  L, 
in  series  with  a  condenser  of  capacity  A,  we  have  the  equation 
E-L-DtC-Q/K=rC,  where  C  =  DtQ.  Show  that  this 
equation  can  be  written  in  the  form 


and  write  down  its  solution  in  the  form  needed  for  practical 
use.  Treat  in  the  same  manner  several  of  the  equations  of 
Section  86. 

371.  Two  circuits,  sl  and  s2,  have  resistances  r1?  r2,  induc 
tances  LI,  L2,  capacities  A1?  K»,  and  a  mutual  inductance  M. 
They  contain  variable  electromotive  forces  Elt  E^  and  carry 
currents  Cl5  Cz.  Show  that  if 


R,  =  r,  -f  L<&  + 
where  <^>  represents  the  operation  Dt, 


If  the  capacities  of  the  circuits  are  negligible,  we  are  to  put 
Rl  =  rx  -}-  Lrf,  J?2  =  ?*2  4-  L.2<^  in  these  results. 
If  Ez  =  0,  Kt  =  oo,  A",  =  oo  ; 


and     C2  =  -  M+E!      1-^2  +  r^  +  r 


474  MISCELLANEOUS   PROBLEMS. 

Write  down  the  solutions  of  these  equations  and  compare 
them  with  the  results  given  in  Section  87. 

372.  Show  that  if  Cv  C2  are  the  currents  in  the  primary  and 
secondary  of  a  transformer,  and  if  the  secondary  circuit  has 
no  capacity  and  contains  no  internal  applied  electromotive 
force,   M<l>Cl+(r2+L2<t>)C2  =  0    or   C2  =  -M^C,/^  +Lrf), 
and  if  Cl  =  Cm  sin  (pt  —  a), 

C2  =     ~    P   m     •  sin  (pt  -  a  +  |  -  tan-lL^/r,). 
V£2y  +  r* 

If,  as  is  often  the  case  in  practice,  i\  is  small  compared  with 
L2p,  we  have,  approximately, 

C2  =  —  MCm  •  sin  (pt  -  a)  -  /L2 
=  M Cm  -  sin  (pt  —  a  —  ?r)  •  /L2. 

In  the  general  case  C2  and  C±  are  not  zero  at  the  same  instant. 

373.  An    electrodynamometer   consists    essentially  of   two 
coils,  one  fixed  and  the  other  movable.     The  movable  coil  is 
furnished  with  an  index  which  moves  on  a  fixed  scale,  and 
the  readings  are  to  be  considered  equal  to  the  product  of  the 
strengths  of  the  steady  currents  Cv  C2  in  the  two  coils.     If, 
however,  these   currents   alternate  rapidly,  the  readings   are 
proportional  to  the  average  value  of  CrC2.     Assuming  that 

J  sin  pt-  sin  qt  •  dt 

=  Sin  O  -?)*•/  2  (p  -q)-  Sin  (p  +  q)t  • /2  (p  +  q), 

\  siii2pt  dt  —  (pt  —  sin^tf  •  cospt)/2p, 

and  that  Cl  =  m  -  sin  pt,  C2  =  n-  sin  qt,  show  that  the  reading 
is  zero  when  p  and  q  are  not  equal.  What  is  it  when  p  =  q  ? 
Plot  a  curve  which  shall  show  the  readings  for  different  values 
of  a,  when  Cl  =  m-  sin  pt}  C2  =  n  •  sin  (pt  —  a),  assuming  that 


sin  pt  •  cospt  dt  =  —  cos  2  pt  •  /4  p. 
If  a  =  %  ir,  the  reading  is  zero. 


MISCELLANEOUS    PROBLEMS.  475 

374.  A  function  z=f(t)  may  be  represented  in  polar 
coordinates  at  any  instant  by  a  point  P,  the  distance  of  which 
from  the  origin  0  is  equal  to  the  numerical  value  of  «,  while 
the  vectorial  angle  XOP  is  equal  to  pt,  where  p  is  any  con 
venient  constant.  The  plane  path  traced  out  by  P  during  any 
time  interval  shows  the  march  of  z  during  the  interval.  If  z 
is  either  a  •  cos  (pt  —  8)  or  a  •  sin  (pt  —  8),  the  path  of  P  is  a 
circumference  of  diameter  a  passing  through  the  origin  :  the 
vectorial  angle  of  the  centre  of  the  circumference  is  8  in  the 
first  case  and  8  -f-  $  TT  in  the  second. 

If  z  is  known  to  be  a  simple  sine  or  a  simple  cosine  func 
tion  of  frequency  p/Zir,  it  is  completely  determined  when  the 
vector  OP0,  which  represents  the  diameter  of  the  circumfer 
ence,  is  given.  If  the  plane  of  the  diagram  were  the  ordinary 
complex  plane,  P0  would  represent  the  complex  quantity 
20  =  x0  +  jym  where  x0  and  yQ  are  the  horizontal  and  vertical 
projections  of  the  diameter  of  the  circumference,  and  j  the 
imaginary  unit ;  and  it  is  often  convenient,  as  Steininetz  has 
shown  in  a  series  of  remarkable  papers,  to  represent  the  har 
monic  function  z  by  the  quantity  ZQ.  With  this  understand 
ing  of  the  meaning  of  the  sign  of  equality,  we  may  write  in 
general  z  =  x0  +  yj:  the  modulus  of  z  is  given  by  the  equa 
tion  |z|  =  Vz02  +  2/02. 

Show  that  if  C'  +j  •  C"  represents  the  current 

C=Cm.sm(pt-$), 

where  tan  8  =  Lp  /  ;•,  in  a  simple  circuit  of  resistance  ?•,  induc 
tance  L,  reactance  x  =  Lp  =  2  -n-nL,  and  impedance  Z\  the 
"  electromotive  force  consumed  by  the  resistance  "  is 

rC=r(C'+j-C"), 

the  "  electromotive  force  produced  by  the  reactance  "  is 
jxC=jxC'-xC", 


476  MISCELLANEOUS    PROBLEMS. 

and  the  electromotive  force  required  to  overcome  the  reactance 
is  xC"  —jxC',  so  that  the  applied  electromotive  force  E  is 

(rC'  +  xC")  +j(rC"  -  xC>)  =  er'  +  ex  -j  =  (r  -  jx)  C, 
and  the  impedance  is  (r  —jx). 

Write  down  an  expression  in  complex  form  for  the  impe 
dance  of  a  simple  circuit  made  up  of  a  number  of  coils  of 
resistances  r±,  r2,  r3,  •  •  •  ,  and  inductances  Ll}  Lz,  L3  in  series. 

Show  that  if  an  electromotive  force,  Em-cospt,  be  applied  to 
a  simple  circuit  of  resistance  r  and  inductance  L  which  con 
tains  a  condenser  of  capacity  k,  the  impedance  Z  has  the  form 
r—j(x  —  x'),  where  x  =  Lp  and  x'  =  \/kp. 

375.  A  long  straight  wire  parallel  to  the  x  axis,  of  resist 
ance  r  and  self-inductance  L  per  unit  length,  is  covered  by  a 
thin  layer  of  insulation  the  outside  of  which  is  kept  at  poten 
tial  zero.  The  capacity  of  the  cable  per  unit  length  is  k,  and 
the  rate  of  leakage  through  the  insulation  of  a  point  where 
the  wire  is  at  potential  V  is  A.  V  per  unit  of  length.  Show 
that  if  C  is  the  current  in  the  wire, 


or,  -' 

DJC  -  Lk-D?C-  (L\  +  rk)  DtC  -  r\C  =  0. 

[Heaviside,  Electrical  Papers,  Vol.  I,  XX  ;  Poincare, 
Comptes  jRendus,  1893  ;  Picard,  Comptes  Rendus,  1894  ; 
Boussinesq,  Comptes  Rendus,  1894  ;  Bedell  and  Crehore, 
Alternating  Currents  ;  Webster,  Electricity  and  Magnetism  ; 
Pupin,  Transactions  of  the  American  Mathematical  Society, 
1900;  The  Electrical  World  and  Engineer,  October,  1901, 
and  February,  1902.] 

376.  Two  circuits  slt  s2,  which  have  self-inductances  Lly  Lz, 
and  a  mutual  inductance  M,  carry  currents  Cly  C2  in  a  magnetic 
field  due  to  these  currents  only.  The  first  circuit,  which  is 
rigid,  contains  a  generator  of  constant  electromotive  force  El  ; 
the  second,  which  is  deformable,  contains  no  generator,  so  that 


MISCELLANEOUS    PROBLEMS.  477 


,  +  MC^/dt  =  rjCi,  -  d(3IC\  +  L,C*)/dt  =  r,C,. 
Show  that  if  one  (.r)  of  the  generalized  coordinates  which 
define  the  conformation  of  the  second  circuit  receives  the 
increment  dx  during  the  time  dt,  so  that  L.2,  M,  C^  C2  are 
changed  while  Zx  remains  constant,  the  work  d  W  done  by  the 
electromagnetic  forces  is  %  C£  •  dL.2  +  C^-  dJft  and  the  change 
dT  in  the  electrokinetic  energy  is 

±  C,2  •  dL,  +  L&-  dC,  +  MC,  •  dC, 

+  MC,  •  dC,  +  dC,  •  dM  +  L2C,  •  dC2. 
Show  that  the  equation  (—  dp2  =  r2(72  •  dt)  yields 
r,Cr  -  dt  +  C,L,  •  dC,  +  C22  •  dL,  -h  MC2  -  rfCL  +  C.C,  •  dM  =  0, 

and  that  the  energy  (Cfrdt  +  C^dp^  furnished  during  the  inter 
val  dt  by  the  generator  in  the  first  circuit  is  equal  to  d  W  +  dT 
plus  the  energy  dissipated  in  heat  in  the  two  circuits.  If  C2 
is  originally  zero,  the  expressions  for  d  W  and  c/Tare  much  sim 
plified.  In  any  given  case  dx  /dt  is  virtually  determined  by  the 
mechanical  equation  of  motion  of  the  moving  parts  of  s2  ;  its 
value  will  evidently  be  greater  or  smaller,  other  things  being 
equal,  according  as  the  electromagnetic  forces  are  assisted  or 
opposed  by  external  forces.  L.2.  J/  are  to  be  regarded  as 
given  functions  of  x  and  other  variables  which  do  not  here 
enter,  and  dL2/dt,  dM/dt  can  be  written  D^L^dx/dt)  and 
D^I^dx/dt).  The  mechanical  equation  and  the  first  two 
equations  of  this  problem  form  a  set  which  completely  deter 
mine  x,  C1?  C.2  as  functions  of  the  time. 

If  a  circuit  s  is  threaded  by  J/4>  lines  from  a  magnetic 
shell  of  strength  3>,E.dt-  d(3f3>  4-  LC)  =  rC  •  dt  and,  in  the 
general  case,  JIT,  3>.  L,  and  C  are  all  functions  of  the  time. 

377.  If  fji  is  the  mass  of  the  slider  AB  in  Fig.  69,  and  if 
a  constant  force  A"  be  applied  to  AB  towards  the  right,  if 
DG  =  I,  GB  =  x,  r  =  Dp,  we  have 

H  .  Dti-  =  i  C--DJ.  +  CHI  +  X, 
E-dt-d(LC+  Hlx)  =  rC-  dt. 


478  MISCELLANEOUS    PROBLEMS. 

Show  that  if  we  can  neglect  the  effect  of  the  field  due  to  the 
current  in  DAJ3G,  and  if  the  change  in  r  during  the  motion  is 
inappreciable, 


C  =  \_E/r  +  X/Hl~\&-K-  XI  HI, 
v  =  [rX/HH*  +  E/HZ]  (1  -  *-"), 

where  A  =  H*l*/nr. 

If  X  is  positive,  the  current  changes  sign  when 
HHH  =  fir  •  log  \_(EHl  +  rX)/rX~\. 

378.  A  condenser  made  of  two  circular  pieces  of  tin  foil, 
each  28.58  centimetres  in  diameter,  separated  by  a  plate  of 
plane  glass  of  inductivity  6,  J  of  a  centimetre  thick,  is  dis 
charged  by  means  of  a  piece  of  non-magnetic  wire  1  metre 
long,  bent  into  the  form  of  a  nearly  complete  circle.     The 
resistance  of  the  circuit  is  0.001  ohm,  and  its  self-induction 
1000    electromagnetic    absolute   units.     Assuming    that    the 
farad  is  equivalent  to  9  x  1011  electrostatic  absolute  units  of 
capacity,  show  that  the  discharge  will  be  oscillatory  with  a 
period   of    about    2.3  x  10  ~7   seconds.     The   time   constant, 
2L  /  R,  is  0.002  second.     The  amplitude  would  be  reduced 
to  YoVu  of  its  initial  value  in  about  0.014  second,  and  to 
T_TJ^_Tr  of  this  value  in  about  0.028  second. 

379.  A  spherical  shell  of  copper  of  small  uniform  thickness 
and  of  radius  a  is  in  a  magnetic  field  of  uniform  intensity  H. 
Show  that  the  work  required  to  withdraw  it  instantaneously 
from  the  field  is  J  H*a*.     [M.  T.] 

380.  An  alternating  electric  current  C  •  cos  pt  is  made  to  flow 
along  a  straight  wire  of  uniform  circular  section.     Prove  that 
the  current  strength  at  a  distance  T  from  the  axis  of  the  wire 
is  given  by  the  real  part  of  Cka-  J0(kr)  •  epti  /  [2  ira?  •  Ji  (ka)], 
where  a  is  the  radius  of  the  wire,  p  its  specific  resistance,  and 
k  =  (1  -  i)  V2~^/  Vp.     [M.  T.] 


MISCELLANEOUS    PROBLEMS.  479 

• 

381.  If  x  is  a  function  continuous  within  the  extremely  short 
time  interval  T,  and  if  T  represents  any  instant  during  the 

interval.    (   x  •  dt  is  small  and    I    dt  \    x-  dt  much  smaller. 

JQ  Jo       */o 

A  wire  circuit  of  conductivity  C  and  self-inductance  L  is 
situated  in  the  field  of  a  magnetic  system  which  undergoes 
a  disturbance  of  impulsive  character  such  that  at  the  end  it 
has  returned  to  its  initial  state.  Prove  that  if  M  denote  the 
change  in  the  induction  through  the  circuit  due  to  the  field  at 
any  instant  during  the  disturbance,  and  if  N  is  the  time  inte 
gral  of  M  taken  throughout  the  whole  time  of  disturbance,  then 
the  induced  current  at  any  subsequent  time  tisNe~t/CL-/  CL2. 
[M.  T.] 

382.  Show  that  if  the  branches  p,  q,  r,  and  5  of  the  Wheat- 
stone  net  have  self-inductances  Lp,  LqJ  Lr,  Ls,  and  contain  con 
densers  which  have  capacities  kp,  kq,  kr,  ks,  and  if  the  current  C 
in  the  main  circuit  is  a  given  function  of  the  time,  the  currents 
in  the  other  branches  are  to  be  found  from  the  equations 


+  Cr/kr  -  Ls  •  DfC.  -  s  •  DtC,  -  Cs/ks  -g.DtCg  =  0. 

(1)  Prove  that  if  Cx'  is  the  current  in  any  branch,  x,  when 
C  =  F(t)t  and  Cx"  the  current   in   the  same   branch  when 
C  =/(*),  Cx'  +  Cx"  will  be  the  current  when  C  =  F(t)  +/(*)• 

(2)  Show  that  if  Cx'  +  i  •  Cx"  is  the  value  of  Cx  obtained 
from  these  equations  when  C  =  F(t)  -f  i-f(t),  Cx'  would  cor 
respond  to  C  =  F(t)  and  Cx"  to  C  =/(*). 

(3)  Show  that  if  C  =  A  •  ext,  the  equations  are   satisfied, 
when  the  coefficients  are  properly  determined,  by  the  values 


480 


MISCELLANEOUS    PROBLEMS. 


and  that  if  bx  =  1  4  Xkx(x  +  X  •  Lx), 

(bpkq  4  *A)  4.  +  0**A4r  =  kM» 

(&  A  +  &A)  A  -  (  *  A  +  &A  +  ?AJfc  A)  A  =  M 

Find  4,. 

(4)  Show  that  if  the  condensers  are  all  removed,  if 

C=A-  eA<, 
and  if 


4  aq)  (ar  4  a,) 


t  sin  mt),  Ag  has  the  form 


4   £  Op    4   aq   +   ar   +   as)~ 

(5)  If  X  =  mi,  C  —  A  (cos  ra£ 
M  +  JVi',  and  Cg  the  form 

(Jf  +  Ni)  (cos  m^  H-  i  sin  mt)  =  (M>  cos  mt  —  N-  sin  m^) 

4-  *(-3f  •  sin  m^  4-  -^-  cos  mf). 

If  the  condensers  are  all  removed,  and  if  C  =  A  •  cos  mt,  what 
is  the  condition  that  no  current  shall  pass  through  g  ? 

383.  Show  that  if  (1)  the 
branches  p,  q,  r,  and  s  of  the 
Wheatstone  net  have  self- 
inductances  LJ},  Lq,  Lr,  Ls,  and 
are  in  parallel  with  branches 
of  negligible  resistance  having 
capacities  kp,  kq,  kr,  k,,  if  (2)  the 
current  C  in  the  main  circuit 
(Fig.  134)  is  a  given  function 
of  the  time,  and  if  (3)  the  cur 
rents  in  the  branch  x  of  the  net  and  in  the  condenser  circuit 
parallel  to  it  are  denoted  by  Cx  and  CJ  respectively,  the  currents 
are  to  be  determined  with  the  help  of  the  equations 


FIG.  134. 


Cr  4  Cr'  +  C,  4-  C.1  =  C,     Cp 


Cp'  = 


Cr  4 


MISCELLANEOUS    PKOBLEMS.  481 


and  !••  Cr  -s.C,  -g.  Cg  =  -Lr-DtCr 

If  the  results  of  applying  the  operations 

[1  +  A-x  (x  .  Dt  +  Lx  -  D,2)],  [x  +  Lx  •  D  J  to  Cx 

be  denoted  by  <f>(Cx),  ^(C>)  respectively,  these  equations  yield 
the  five  equations  which  follow  : 


Show  that  if  C  =  A  •  e**,  these  equations  are  satisfied,  when  the 
coefficients  are  properly  determined,  by  the  values  Cp  =  Ap  -  e", 
Cq  =  Aq.  <P,  Cr  =  A,.  •  eA',  Cs  =  As  •  e",  Cg  =  Av-  e",  and  that  if 
ax  =  x  +Lx-\,  bx  =  1  +  k,\(x  +£,-  A), 


If  A  is  real,  and  if  X  =  mi,  where  m  is  real, 
C  =  A  (cos  mt  +  /  •  sin  mt), 
Ag  is  of  the  form  M  -f-  Ni,  and  Cg  of  the  form 

(M+  Ni)  (cos  mt  +  i  •  sin  mt) 
or  (JIf  •  cos  mt  —  N-  sin  ?w#)  -f-  i  (M-  sin  ??i#  +  jVcos  mt). 

The  real  part  of  Cg  is  of  the  form  ^/  M~  -\-  N2  •  cos  (mt  -  8), 
where  tan  8  =  —  N/M.  Prove  that  this  would  be  the  value  of 
Cg  if  the  value  of  C  were  A  •  cos  /«#. 

Show  that  if  C  =  A-  cos  mt,  if  the  condensers  are  all 
removed,  and  if  Lq  =  L^  =  0,  Cg  will  be  zero  for  all  values 
of  m  if  X^/Zr  =  fj/s  =p/r. 

Prove  that  if  C  =  A  -  cos  mt,  if  the  inductances  are  neg 
ligible,  and  if  A*,  =  A*.  =  0,  Cg  will  be  zero  for  all  values  of  m 


482  MISCELLANEOUS    PROBLEMS. 

Prove  that  if  all  the  L's  and  &'s  except  kp  and  Ls  are  zero, 
Cg  will  be  zero  if,  and  only  if,  Ls  =  qrkp  =  pskp. 
Show  how  to  find  the  value  of  Cg  if  C  =  A  •  sin  mt. 

384.  An  infinite  mass  of  metal  has  one  plane  face,  which 
is  the  yz  plane.     At  the  time  zero,  uniform  currents  parallel 
to  the  z  axis  are  induced  in  the  plane  by  a  sudden  change  in 
the  magnetic  field,  which  after  the  change  remains  constant. 
It  is  evident  that  u  and  v  will  remain  zero  and  that  w  is  a 
function  of  x  only,  so  that  (Section  88)  4  ?r/xX  •  Dtw  =  D^w  and 
w  =  A  •  eu/Fj  where  u  =  —  Tr/xAx2  / 't.     Show  that  w  will  have 
its  maximum  value  at  a  distance  x0  from  the  plane  face  at  a 
time  t  =  2  ir^Xx^  and  that  this  value  is  A  /(x  V2  Tr/uAe).     When 
t  is  large,  w  has  nearly  the  same  value  for  all  moderate  values 
of  x.     Assuming  that  for  copper  /xA  =  1/1600  and  for  iron 
1/10,  show  that  the  maximum  current  will  be  attained  at  a 
depth  of  16  centimetres  in  copper  and  1.26  centimetres  in 
iron  after  1  second. 

385.  An  infinite  conductor  has  one  plane  face  (the  yz  plane), 
but  is  otherwise  unlimited.     Periodic  currents  parallel  to  the 
z  axis  and  independent  in  intensity  of  y  and  z  are  induced  in 
this  conductor  by  some  cause  on  the  negative  side  of  the  yz 
plane.     Since  u  and  v  are  zero,  w  must  (Section  88)  satisfy  the 
equation  4  TT\(JL  •  Dtw  =Dxzw,  and  of  this  equation   Celt  •  e~nx, 
where  I  =  kH/^icX^  n2  =  kH,  n  =  k(l  +  i)/^/2,  is  a  special 
solution.   The  real  part  of  a  complex  solution  of  this  equation  is 
itself  a  special  solution ;  and  if  we  assume  that  w  =  Cm  cos  pt, 
when  x  =  0,  we  learn  that 

w  =  Cme~*V2n^p .  cos  (pt  -  x  V2  7rA/xp), 

a  simple  harmonic  expression  the  amplitude  of  which  is 
Cme~2nxV^f,  where /is  the  number  of  alternations  per  second. 
Show  that  this  amplitude  has  I/ rath  of  its  surface  value  at 
the  depth  log  w/(2ir  VXf/).  Show  that  if  the  frequency  is 
100,  the  amplitudes  1  centimetre  from  the  surface  will  be 
0.208  Cm  in  the  case  of  copper,  and  0.0000000024  Cm  if  the 


MISCELLANEOUS    PROBLEMS.  483 

conductor  is  made  of  iron.  Show  also  that  if  the  frequency  is 
10,000  the  amplitude  in  the  case  of  copper  at  a  point  1  centi 
metre  from  the  surface  would  be  0.00000015  Cn. 

386.  Show  that  if  Uy  F,  W  are  the  components  of  a  vector 
taken  at  every  point  in  the  direction  in  which  the  orthogonal 
curvilinear  coordinates  u,  v,  u-  increase  most  rapidly,  the 
components  of  the  curl  of  the  vector  are 


A  very  interesting  proof  of  Stokes's  Theorem  in  curvilinear 
coordinates  has  been  given  by  Prof.  A.  G.  Webster  in  the 
Bulletin  of  the  American  Mathematical  Society  for  1898. 

387.  The  whole  induction  flux  through  a  linear  circuit,  sk, 
is  equal  to  the  line  integral,  7t,  of  the  tangential  component 
of  a  vector  potential  of  the  induction  taken  around  the 
circuit,  so  that  the  electrokinetic  energy,  T,  of  a  set  of 
circuits  5lf  s.2,  s3,  •  •  •,  which  carry  currents  (715  (72,  C3,  •  •  -,  is 


Cklk  =  ±       C*lF*  •  cos  (x,  5,)  +  Fv  -  cos  (y, 


If  the  linear  circuits  are  the  filaments  of  a  massive  conductor 
in  which  the  components  of  the  current  are  it,  r,  w,  and  if  Sk  is 
the  area,  at  any  place,  of  the  cross-section  of  the  filament  sk  ; 
Ck  •  cos  (x,  sk)  =  u  •  Sk,  Ck  •  cos  (y,  sk)  =  v  •  Sk,  Ck  •  cos  (2,  st)  =  w  -  Sk. 
Show  that  we  may  write 


where  the  integration  is  to  be  extended  over  all  space  where 
currents  exist. 


ANSWERS. 


ADDITIONAL   ANSWERS   TO  PROBLEMS. 
CHAPTER   I. 

d\  x—  113Q32  Y—  96501  7?  — o^ 

~  274625'    "  274625'  *  +  ' 

.0  =  40°  29'+. 

(2)  R  =  m\,  cosa=— ,  cos/3  =  — -,  cosv  =  — -,  where 

crA                 tr\  cr\. 


(5)  If  2  Z  be  the  length  of  the  wire,  and  if  the  axis  of  the 
wire  be  taken  for  the  axis  of  abscissas  with  the  origin  at  the 
middle  point,  the  required  equation  is 


(6)  If  the  radius  of  the  earth  be  taken  as  3960  miles,  and 
the  mass  of  a  cubic  foot  of  water  as  62.5,  one  poundal  is  equiv 
alent  to  952  million  attraction  units  approximately. 

(8)  If  c  be  the  distance  of  the  point  P  from  the  centre  of  the 

sphere,   the   required   attraction   is  m\ ,  if  P  is 

[c2      8(c±r)2J 

without  the  solid  ;  -^,  if  P  is  within  the  cavity,  and 

8r'2 


m 


if  P  is  a  point  in  the  mass  of  the  solid. 

(9)  The  attraction  of  a  hemisphere  of  radius  R  at  a  point  P 
facing  the  flat  side  of  the  hemisphere  and  lying  on  the  perpen 
dicular  to  this  side  erected  at  the  centre  0  is 

484 


ANSWERS.  485 

M 


-  2  a3  +  (2  a2  -  ^2)  V.ft2  +  a2] ,  where  a  =  OP. 

The  attraction  of  the  other  hemisphere  ma}'  be  found  by  sub 
tracting  this  quantity  from  the  attraction  due  to  the  whole 
sphere  of  which  this  hemisphere  is  a  part.  See  §  9. 

(10)  (a)  That  the  density  varies  inversely  as  the  distance 
from  the  centre. 

W       _^ir?+^ i_i 

ivJPLr     981itr      327?iJ* 

(12)  Here  d  is  supposed  to  be  greater  than  a. 

(15)  The  attraction  is  34.9,  and  its  line  of  action  makes  an 
angle  of  1°  49 'with  the  line  joining  the  centre  of  the  sphere 
with  the  point  in  question. 


CHAPTER   II. 

(7)  That  the  force  is  constant. 
(9)   Fo.^ 


_ 

3 

jr         2      /-_      14       .A      T^      2      40 
F2_3  =  -7J7>   2/  ---  c2   ;    Fi_.-«p— 

3       \  c  /  o        c 

(11)  Yes;  1.46  -. 


ifr>5,   F=^l°2. 
r 

(15)   No. 

(20)    (1)  About  1,830,000  tons  of  2000  pounds  each. 


INDEX. 


Activity,  316,  465. 

Alternate    currents    in    inductive 

circuits,  312,  329,  464-482. 
Ampere,  265. 
Ampere,  the,  233,  298. 
Apparent  charges,  183. 
Apparent  electromotive  forces,  317. 
Attraction. 

centrobaric  bodies,  362,  371. 

cones,  8,  25,  349. 

curved  wires,  25,  345. 

cylinders,  7,  26,  337,  346,  347, 
348,  358,  368,  369,  376,  377. 

cylindrical  distributions,  60,  72. 

cylindrical  shells,  26,  349. 

discrete  particles,  2,  25. 

ellipsoidal  homoeoids,  16. 

ellipsoidal  surface  distributions, 
141,  160,  378. 

elliptic  cylinders,  376,  377,  379. 

focaloids,  378. 

given  mass,  19,  350. 

hemispheres,  13,  15,  26,  352,  353, 
35«r\ 

hollow  spWes,  26,  27. 

paraboloids,  28,  353. 

similar  solids,  350. 

solid  ellipsoids,  117,  378. 

special  laws  of,  351,  359,  361. 

spheres,  13,  18,  26,  27,  350,  358. 

spherical  distributions,    56,   72, 
368,  375. 

spherical  sectors,  351. 

spherical  segments,  27,  352. 


Attraction, 
spherical  shells,  10,  11,  18,  26, 

27,  35,  58,  350,  352. 
spheroids,  28,  380,  381. 
straight  wires,  3,  4,  ,25>_2fir_34, 

71,  344,  345,  346,  359. 
thin  plates,  22,  28,  347,  348,  349. 
two  rigid  bodies,  24,  350,  353. 
two  spheres,  23,  337,  338,  339. 
two  wires,  23,  73,  344. 

Ballistic  galvanometers,  442. 
Bocher,  371. 

Capacity,  159,  161,  307,  309,  321. 
(See  Condensers.) 

Centrobaric  distribution,  362,  371. 

Charged  conductors,  146,  148,  157, 
167,  171,  385,  386,  387,  388, 
389,  390,  393,  394,  395,  396, 
397,  402,  403,  410. 

Coefficients  of  potential,  induction, 
and  capacity,  157,  390,  397. 

Columnar  coordinates,  63, 141,  354, 
421. 

Condensers,  161,  164,  166,  176, 
184,  307,  388,  389,  390,  393, 
396,  401,  402,  409,  413,  418, 
419,  420,  439,  440,  441,  466, 
467,  478,  479,  480. 

Conditions  which  determine  func 
tions,  104,  107,  133,  134,  135, 
136,  137,  138,  180,  203,  245, 
371,  382,  415. 


486 


INDEX. 


487 


Conductivity,  227. 
Cones,  8,  25,  349. 
Conjugate  functions,  412,  429,  430, 

431,  432,  433. 
Convergence,  111,  138. 
Coulomb,  the,  233. 
Coulomb's  Equation,  89,  130,  179, 

183. 

Curl,  111,  138, 139, 143, 382,383,483. 
Current  induction,  291. 
Currents  in  cables,  428,  429,  437. 
Curvilinear   coordinates,   65,    136, 

137, 141, 182, 384, 385, 421, 483. 
Cylinders,  7,  26,  346,  347,  348,  358, 

369,  376,  377. 
Cylindrical   distributions,   60,   72, 

368,  369,  371,  375,  404,  408, 

409,  410. 

Darwin,  364. 

Depolarizing  force,  206,  379. 

Derivatives  of  the  potential  func 
tion,  30,  31,  32,  36,  40,  45,  50, 
72,  73,  91,  360,  361,  366. 

Derivatives  of  scalar  functions,  115, 
116,  138,  382. 

Dielectrics,  146,  176,  199,  413,  414, 
415,  416,  418. 

Dimensions  of  physical  quantities, 
210,  338,  462. 

Dirichlet,  50,  104,  125. 

Displacement  currents,  335. 

Dissipation  function,  240. 

Divergence,  111,  138,  141,  382, 
383,  384. 

Divided  circuits,  235. 

Double  layers,  144,  214. 

Doublets,  196,  434,  436. 

Effective  electromotive  forces,  317. 
Electrical  displacement,  177. 


Electrical   images,   167,   170,   400, 

401,  404,  410,  411. 
Electrical  intensity,  177. 
Electrodynamic  potential,  273,  275. 
Electrodynamics,    262,    267,   271, 

273,  276,  297,  454-459. 
Electrodynamoineter,  474. 
Electrokinematic  equilibrium,  222, 

241,  245,  246. 

Electrokinematics,  222,  246. 
Electrokinetic    energy,    271,   281, 

296,  461,  465,  477,  483. 
Electrokinetic  momentum,  297,465. 
Electromagnetic  fields  due  to  closed 

linear  circuits,  259,  454,  455, 

456-459. 
Electromagnetic     fields     due     to 

straight    currents,    251,    255, 

454. 

Electromagnetic  units,  233,  298. 
Electromagnetism,  251. 
Electromotive  force,  230. 
Electromotive   force,    triangle    of, 

316,  319,  320,  321. 
Electrostatic     potential    functions 

within  conductors  which  carry 

currents,  241,  246. 
Electrostatic  units,  233. 
Electrostatics,  145. 
Ellipsoidal   conductors,    160,  401, 

402. 

Ellipsoidal  homceoids,  16. 
Ellipsoidal  shells,  16,  378. 
Ellipsoidal  surface  distributions, 

141,  378. 

Ellipsoids,  117,  378,  380. 
Elliptic  cylinders,  376,  377,  379. 
Energy,  43,  97,  183,  269,  280,  364, 

368,  391,  401,  418,  453,  454. 
Energy  of  charged  conductors,  171, 

175,  183,  391,  394,  401,  418. 


488 


INDEX. 


Equations  of   the  electromagnetic 

field,  332. 
Equilibrium  of  fluids,  70,  73,  74, 

365,  381. 
Equipotential  surfaces,  37,  71,  72, 

73,    122,   141,    357,    369,    371, 

372,  403. 
Ewing,  291. 

Farad,  the,  233. 

Faraday's  disc,  272,  298. 

Faraday  tubes,  152. 

Field  components,  21,  30,  177,  255, 

265,  283,  332. 
Fleming,  291. 
Flow  of  force,  151,  365. 
Focaloids,  378. 

Galvanometers,  455,  456,  457,  458. 
Gauss's  Theorem,  52,  66,  78,  129, 

151,  177. 
Gibbs,  221. 
Gradients,  115,  137,  138,  384,  385, 

421,  483. 
Gravitation,  1. 
Gravitation  constant,  2,  370. 
Gravity,  15,  342,  343,  344,  347,  370. 
Gray,  269. 

Green's  distribution,  109,  446. 
Green's  Function,  384. 
Green's  Theorem,  91,  129,  384. 

Hard  and  soft  media,  202,  204,  208. 
Harmonic  Functions,  45,  100,  103, 

104,  105, 135, 137, 143, 382, 415. 
Heat   developed  in  circuits  which 

carry  currents,  238,  272,  297. 
Heaviside,  221,  282,  402. 
Helmholtz,  221. 

Hemispheres,  13,15, 26, 352, 353, 358. 
Hilbert,  105. 


Hollow  conductors,  152. 
Hysteresis,  289,  460. 

Impedance,  315,  318,  465,  476. 
Induced  charges,  146,  153, 156,  167, 

184,  201,  394,  395. 
Induced  currents,  292,  298,  299, 

302,  326,  463,  464. 
Induced  electromotive  force,  292, 

295,  297,  302. 

Induced  polarization,  203,  204,  207. 
Inductance,  278,  458,  461,  462,  476. 
Induction,  146,  152. 
Induction  coils,  326. 
Induction  flux,  151,  260,  292. 
Induction  vector,  178,  201,  202. 
Inductivity,  176,  200. 
Intrinsic  charges,  183. 
Intrinsic  energy  of  a  distribution, 

the,  43,  97,  183. 
Inversion,  397,  399,  400,  401,  411. 

Joule,  the,  239. 

Kelvin,  105,  206. 
Kirchhoff,  234,  280. 
Kirchhoff's  laws,  234. 

Laplace's    Equation,    44,    65,    71, 

245,  357,  359,  360. 
Laplace's  Law,  262. 
Law  of  gravitation,  1. 
Law  of  nature,  75,  145. 
Linear  conductors,  226,  230,  421. 
Linear  differential  equations,  302 

304,  307,  309,  311,   314,   470, 

471,  472,  473,  474. 
Lines  and  level  surfaces  of  vectors, 

112,  123,  140,  382. 
Lines  and  surfaces  of  flow,  243, 

246,  247,  249. 


INDEX. 


489 


Lines  and  tubes  of  force,  55,  71,  72, 
73,  122,  150,  187,  188,  251,  260, 
288, 359, 367, 372, 373, 391, 394. 

Logarithmic  potential  functions, 
126,  385,  406,  407. 

Magnetic  energy,  269. 
Magnetic  induction,  260,  287. 
Magnetic  lines,  391. 
Magnetomotive  force.  287. 
Magnets,  199,  442,  443,  444,  445, 

446,  450,  451,  453. 
Maximum  and  minimum  theorems, 

103,  135,  136,  240. 
Maxwell,  334. 

Maxwell's  Current  Equations,  281. 
Mechanical  action  on  a  conductor 

which  carries  a  current  in  a 

magnetic  field,  262,  264,  267. 
Motion  under  gravitation,  71,  338, 

339,  340,  341-344. 
Mutual  energy,  42,  269,  273,  276, 

368,  395,  401. 
Mutual  energy  of  distribution  and 

field.  451,  452,  453,  454. 
Mutual  inductance  of  two  circuits, 

276,  278. 

Neumann,  273. 

Newton,  349. 

Non-homogeneous  conductors,  244, 

245. 
Normal  force,  89. 

Ohm,  the,  233,  298. 
Ohm's  Law,  227. 

Paraboloids,  28,  353. 
Pendulums,  26,  342. 
Permeability.  176. 
Perry,  469,  472. 


Planetary  motion,  341. 

Poisson,  334,  446. 

Poisson's  Equation,  61,  66,  79,  129, 

147,178,182,201,202,360,421. 
Poisson's  Integrals,  102,  132. 
Polarization,  185,  192,  198. 
Polarization  moments,  186,  193. 
Polarization  vector,  186,  194. 
Polarized  cylinders,  448,  449. 
Polarized  ellipsoid,  189,  207. 
Polarized  shells,  214,  450. 
Polarized  spheres,  187,  447. 
Potential  difference,  231,  318,  422, 

466. 

Potential  function 
as  measure  of  work  and  energy, 

41,  78. 

average  value  on  spherical  sur 
face,  67. 

definition,  29,  354. 
derivatives  of,  30,  31,  32,  36,  40, 

44,  45,  50,  61,  72,  73,  89,  91, 

130, 179, 183,360,  361,365, 366. 
properties  and  characteristics  of, 

32,  40,  44,  67,  68,  78,  80,  86, 

107,  179. 
special  cases,  34,  35,  36,  58,  60, 

71,   72,  74,  80,  82,   125,   197, 

355-365,  375. 
Poynting,  371. 
Pupin,  467,  476. 

Rayleigh,  307. 

Eeactance,  315. 

Real  charges,  183. 

Reluctance,  287. 

Repelling  matter,  75,  70. 

Resistance,  227,  247,  249,  422,  426, 

433. 

Resonance,  323,  467. 
Ring  magnets,  286. 


490 


INDEX. 


Self-inductance,  278,  296,  300-331. 
Solenoidal   and   lamellar    vectors, 

111,  138,  139,  140,  143,  144, 

221,  382,  383. 
Solenoidal  polarization,  198,  203, 

204,  449,  450. 
Solenoids,  284. 
Solid  angles,  11,  49,  53,  215,  261, 

349. 

Space  derivatives  of  scalar  func 
tions,  115,  138,  382. 
Specific  inductive  capacity,  176. 
Spheres,  13, 18,  23,  26,  27,  350,  358. 
Spherical  condensers,  161, 184,  389, 

390,  413,  419. 
Spherical  conductors,  159, 161, 167, 

394,  396,  401-403. 
Spherical  coordinates,  63,  384,  421. 
Spherical  distributions,  56,  72,  184, 

210,  368,  372,  375. 
Spherical  segments,  27,  352. 
Spherical  shells,  10,  11,  18,  27,  35, 

58,  350,  360,  375. 
Spheroids,  28,  144,  370,  379,  380. 
Steady  currents  in  linear  circuits, 

222-241, 421-429, 441, 454-459. 
Steinmetz,  475. 
Stokes's  flux  function,  367. 
Stokes's  Theorem,   113,  219,  252, 

282,  295,  332,  383,  483. 
Strength  of  field,  2,  147. 
Superficial  induced  currents,  299, 

479,  482. 
Surface  distributions,   83,   85,  88, 

109,  146-176,  385-420. 
Surface  pressure,  90. 
Susceptibility,  200,  281. 

Theorems    involving   surface   and 
volume  integrals,  47,  54,  66, 


93,  94,  95,  97,  98,  100,  101, 
102,  103,  104,  113,  132,  135, 
136, 137, 144, 220, 356, 357, 384, 
414,  415,  452,  453,  454,  460. 

Thomson,  J.  J.,  299. 

Thomson's  Theorem,  104. 

Tide-generating  forces,  363,  364. 

Transformers,  331,  474. 

Triangle  of  resistances,  316. 

Two-fluid  theory,  145. 

Uniform  polarization,  186,  188. 
Uniformly  polarized  distributions, 

186,  189,  205,  207. 
Units,  233,  298,  462. 
Units  of  force,  2,  25,  31,  80,  210, 

337,  338,  462. 

Variable  currents,  423,  437,  439, 
440,  441. 

Variable  currents  in  inductive  cir 
cuits,  301,  326. 

Vector  lines  and  surfaces,  112, 123. 
140,  382. 

Vector  potential  functions,  112, 
139,  140,  218,  294,  452,  453. 

Vector  product,  293. 

Vectors,  14,  111,  139. 

Volta's  Law  of  Tensions,  229. 

Volt,  the,  233,  298. 

Webster,  206,  265,  483. 
Wheatstone's  net,  236,  241,  305, 

306,  311,  427,  479,  480. 
Wires  or  rods,  3,  4,  23,  25,  26,  34, 

71,  73,  344,  345,  346,  359. 
Woodward,  370. 
Work,  41,  78,  354,  355,  401. 

Zonal  harmonics,  261, 373, 374, 375. 


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student  is  taught  by  the  method  of  translation  of  the  origin 
to  handle  any  equation  of  the  second  degree  in  which  the 
x  y  term  does  not  appear.  In  particular,  the  equations  of 
the  tangent,  the  normal,  and  the  polar  have  been  determined 
for  such  an  equation.  Only  later  is  the  general  equation  of 
the  second  degree  fully  discussed. 

In  the  solid  geometry,  besides  the  plane  and  the  straight 
line,  the  cylinders  and  the  surfaces  of  revolution  have  been 
noticed,  and  all  the  quadric  surfaces  have  been  studied  from 
their  simplest  equations.  This  study  includes  the  treatment 
of  tangent,  polar,  and  diametral  planes,  conjugate  diameters, 
circular  sections,  and  rectilinear  generators. 

Throughout  the  work  no  use  is  made  of  determinants  or 
calculus. 

GINN  &  COMPANY,  PUBLISHERS, 

BOSTON.        NEW  YORK.        CHICAGO. 


WENTWORTH'S 

New  Plane  and  Spherical 

Trigonometry,  Surveying,  and  Navigation 

By  GEORGE   A.  WENTWORTH 


Half  morocco.    412  pages.    For  introduction,  $1.20 


IN  this  book  the  principles  have  been  unfolded  with  the 
utmost  brevity  consistent  with  simplicity  and  clearness, 
and  interesting  problems  have  been  introduced  with  a  view 
to  awaken  a  real  love  for  the  study.  Much  time  and  labor 
have  been  spent  in  devising  the  simplest  proofs  for  the  propo 
sitions,  and  in  exhibiting  the  best  methods  of  arranging  the 
logarithmic  work.  Answers  are  included. 

The  special  features  of  the  New  Plane  Trigonometry 
are  sufficient  practice  in  the  use  of  the  radian  as  the  unit 
of  angular  measure,  the  solution  of  simple  trigonometric 
equations,  the  solution  of  right  triangles  without  logarithms, 
a  brief  treatment  of  anti-trigonometric  functions,  and  a 
chapter  on  the  development  of  functions  of  angles  in  infinite 
series.  It  also  contains  the  latest  entrance  examination 
papers  of  some  of  the  leading  colleges  and  scientific  schools, 
and  a  large  number  of  miscellaneous  problems  in  trigonom 
etry  and  goniometry. 

Teachers  can  omit,  at  their  discretion,  the  chapter  on 
construction  of  tables,  and  many  of  the  miscellaneous 
problems  in  trigonometry  and  goniometry. 

The  Spherical  Trigonometry,  Surveying,  and  Navigation 
has  been  entirely  rewritten,  and  such  changes  made  as  the 
most  recent  data  and  methods  required. 


GINN    Sc   COMPANY,  Publishers 

Boston  New   York  Chicago  San   Francisco 

Atlanta  Dallas  Columbus  London 


WENTWORTH'S   GEOMETRY 

REVISED. 
BY  GEORGE  A.  WENTWORTH. 

Wentworth's  Plane  and  Solid  Geometry.    Revised.    473  pages. 

Illustrated.     For  introduction,  $1.25. 

Wentworth's  Plane  Geometry.     Revised.     256  pages.    Illustrated. 
For  introduction,  75  cents. 

Wentworth's  Solid  Geometry.    Revised.     229  pages.    Illustrated. 
For  introduction,  75  cents. 

THE  history  of  Wentworth's  Geometry  is  a  study  in 
evolution.  It  was  the  corner  stone  on  which  was  built 
a  now  famous  mathematical  series.  Its  arrangement  and 
plan  have  always  appealed  to  the  eager  student  as  well 
as  to  the  careful  teacher.  It  was  the  first  to  advocate  the 
doing  of  original  exercises  by  the  pupils  to  give  them  inde 
pendence  and  clear  thinking.  As  occasion  has  offered, 
Professor  Wentworth  has  revised  the  book  in  the  constant 
endeavor  to  improve  it  and  move  it  a  little  nearer  the  ideal. 
The  present  edition  is  a  close  approach  to  this  end.  It 
represents  the  consensus  of  opinion  of  the  leading  mathe 
matical  teachers  of  the  country.  It  stands  for  exact  schol 
arship,  great  thoroughness,  and  the  highest  utility  to  both 
the  student  and  the  teacher. 

In  this  new  edition  different  kinds  of  lines  are  used 
in  the  figures,  to  indicate  given,  resulting,  and  auxiliary 
lines.  These  render  the  figures  much  clearer.  In  the  Solid 
Geometry  finely  engraved  woodcuts  of  actual  solids  have 
been  inserted,  for  the  purpose  of  aiding  the  pupil  in  visu 
alization.  They  give  just  the  necessary  assistance.  The 
treatment  of  the  Theory  of  Limits  is  believed  to  be  the  best 
presentation  of  the  subject  in  any  elementary  geometry. 


GINN   &   COMPANY,  Publishers, 

Boston.      New  York.      Chicago.      San  Francisco.     Atlanta.     Dallas. 
Columbus.      London. 


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